affil0affil0affiliationtext: Institut de Mathématiques, Université de
Neuchâtel, Switzerland

# Quasi optimal anticodes: structure and invariants

Elisa Gorla and Cristina Landolina

###### Abstract

It is well-known that the dimension of optimal anticodes in the rank-metric is
divisible by the maximum $m$ between the number of rows and columns of the
matrices. Moreover, for a fixed $k$ divisible by $m$, optimal rank-metric
anticodes are the codes with least maximum rank, among those of dimension $k$.
In this paper, we study the family of rank-metric codes whose dimension is not
divisible by $m$ and whose maximum rank is the least possible for codes of
that dimension, according to the Anticode bound. As these are not optimal
anticodes, we call them quasi optimal anticodes (qOACs). In addition, we call
dually qOAC a qOAC whose dual is also a qOAC. We describe explicitly the
structure of dually qOACs and compute their weight distributions, generalized
weights, and associated $q$-polymatroids.

## Introduction and motivation

Rank-metric codes are linear spaces of matrices of given size, with entries in
a finite field $\mathbb{F}_{q}$. The set of $n\times m$ matrices over a field
is a metric space when endowed with the rank metric: The distance between two
matrices is the rank of their difference. Rank-metric codes were introduced by
Delsarte in [5]. Gabidulin [8] and Roth [15] later independently rediscovered
the family of rank-metric codes, which are linear over the field extension
$\mathbb{F}_{q^{m}}$. The study of rank-metric codes is motivated by their
applications to linear network coding, code-based cryptography, and
distributed storage.

The mathematical theory of rank-metric codes includes the study of code
invariants and their bounds. Beyond the size of its elements, also called
codewords, and its dimension, two basic invariants of a rank-metric code are
its minimum distance and maximum rank. The minimum distance of a nonzero rank-
metric code is the least rank of a nonzero matrix in the code, while its
maximum rank is the maximum rank of an element of the code. The Singleton
bound involves the dimension of a rank-metric code and its minimum distance,
while the Anticode bound involves the dimension of a code and its maximum
rank. Codes which meet the Singleton bound go under the name of Maximum Rank
Distance codes (MRD codes) and were first studied by Delsarte in [5]. Codes
attaining the Anticode bound were studied by Meshulam in [12], with a
different motivation and terminology. Within coding theory, they are known as
optimal anticodes. While MRD codes are of direct applied interest, optimal
anticodes are mostly of theoretical interest. For example, they can be used to
define the generalized weights, a family of code invariants. This was done in
[13], where the properties of optimal anticodes are studied.

For a fixed minimum distance $d$, MRD codes are the rank-metric codes which
have minimum distance $d$ and the largest dimension according to the Singleton
bound. Conversely, one could fix a dimension $k$ and ask what are the codes of
dimension $k$ and largest minimum distance. It turns out that, if $k$ is
divisible by the maximum between $n$ and $m$, these are the MRD codes. Else,
they are a different family of codes, called quasi MRD codes. This last family
of codes was introduced and studied in [2].

Something similar happens for the Anticode bound: The optimal anticodes are
the rank-metric codes of largest dimension, among those with a fixed maximum
rank. Similarly to the case of MRD codes, the dimension of optimal anticodes
is divisible by the maximum between $n$ and $m$, say $m$. Moreover, for a
fixed $k$ divisible by $m$, optimal anticodes are the codes with least maximum
rank, among those of dimension $k$. In this paper, we study the family of
rank-metric codes whose dimension is not divisible by $m$ and whose maximum
rank is the least possible for codes of that dimension, according to the
Anticode bound. As these are not optimal anticodes, we call them quasi optimal
anticodes (qOACs).

The paper is organized as follows: In Section 2 we define qOACs and dually
qOACs. We give a complete structural classification of dually qOACs and a
partial one of qOACs. In Section 3 we study the generalized weights of dually
qOACs and of a family of qOACs. In Section 4 we study their the weight
distribution, while in Section 5 we compute the associated $q$-polymatroids.

##  1 Preliminaries

In this section we give an introduction to rank-metric codes and state some
results that we will need throughout the paper. Let $q$ a prime power and
$\mathbb{F}_{q}^{\,n\times m}$ the linear space of $n\times m$ matrices with
entries in the finite field $\mathbb{F}_{q}$. Up to transposition, we may
assume without loss of generality that $n\leq m$. Throughout the paper, we
denote by $e_{1},\ldots,e_{\ell}$ the canonical basis of
$\mathbb{F}_{q}^{\ell}$, i.e., $e_{i}$ is the vector whose coordinates are all
zero, except for a one in position $i$.

######  Definition 1.1.

The rank distance of $A,B\in\mathbb{F}_{q}^{\,n\times m}$ is given by the
function

$\displaystyle d:\mathbb{F}_{q}^{\,n\times m}\times\mathbb{F}_{q}^{\,n\times
m}$ $\displaystyle\longrightarrow\mathbb{N}$ $\displaystyle(A,B)\qquad$
$\displaystyle\longmapsto\mbox{rank}(A-B).$

A rank-metric code $\mathcal{C}$ is a $\mathbb{F}_{q}$-linear subspace of
$\mathbb{F}_{q}^{\,n\times m}$ equipped with the rank distance. The dimension
of $\mathcal{C}$ over $\mathbb{F}_{q}$ is denoted by $\dim(\mathcal{C})$.
Further, we let
$\mbox{d}(\mathcal{C})=\min\\{\mbox{rank}(A):A\in\mathcal{C},A\neq 0\\}$ be
the minimum distance of a nonzero rank-metric code $\mathcal{C}$. In a similar
way, the maximum rank is given by
$\mbox{maxrk}(\mathcal{C})=\max\\{\mbox{rank}(A):A\in\mathcal{C}\\}.$

Two rank-metric codes are equivalent if there is a linear rank-preserving
homomorphism mapping one code into the other.

######  Definition 1.2.

An $\mathbb{F}_{q}$-linear isometry $\varphi$ of $\mathbb{F}_{q}^{\,n\times
m}$ is an $\mathbb{F}_{q}$-linear rank-preserving homomorphism, i.e.
$\mbox{rank}(\varphi(A))=\mbox{rank}(A)$ for all
$A\in\mathbb{F}_{q}^{\,n\times m}$. Let
$\mbox{Isom}_{\mathbb{F}_{q}}(\mathbb{F}_{q}^{\,n\times m})$ be the collection
of $\mathbb{F}_{q}$-linear isometries of $\mathbb{F}_{q}^{\,n\times m}$. Two
rank-metric codes $\mathcal{C},\mathcal{D}\subseteq\mathbb{F}_{q}^{\,n\times
m}$ are equivalent if there is
$\varphi\in\mbox{Isom}_{\mathbb{F}_{q}}(\mathbb{F}_{q}^{\,n\times m})$ such
that $\mathcal{C}=\varphi(\mathcal{D})$. We denote the equivalence by
$\mathcal{C}\sim\mathcal{D}$.

Linear isometries of $\mathbb{F}_{q}^{\,n\times m}$ are classified in [10] for
odd characteristic and in [17] for characteristic equal to 2.

######  Theorem 1.3.

Let $\varphi\in\mbox{Isom}_{\mathbb{F}_{q}}(\mathbb{F}_{q}^{\,n\times m})$.
Then

  * (a)

if $n\neq m$, there exist $N\in\mbox{GL}_{n}(\mathbb{F}_{q})$ and
$M\in\mbox{GL}_{m}(\mathbb{F}_{q})$ such that $\varphi(A)=NAM$ for all
$A\in\mathbb{F}_{q}^{\,n\times m}$.

  * (b)

if $n=m$, there exist $N,M\in\mbox{GL}_{n}(\mathbb{F}_{q})$ such that
$\varphi(A)=NAM$ for all $A\in\mathbb{F}_{q}^{\,n\times m}$, or
$\varphi(A)=NA^{t}M$ for all $A\in\mathbb{F}_{q}^{\,n\times m}$.

Further we define the dual of a rank-metric code $\mathcal{C}$ using the
standard scalar product for matrices.

######  Definition 1.4.

Let $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ be a rank-metric code
and let $\mbox{Tr}(A)$ denote the trace of a square matrix $A$. The dual of
$\mathcal{C}$ is defined as

$\mathcal{C}^{\perp}=\\{M\in\mathbb{F}_{q}^{\,n\times
m}:\mbox{Tr}(MN^{t})=0\mbox{ for all }N\in\mathcal{C}\\}.$

For a matrix $M\in\mathbb{F}_{q}^{\,n\times m}$
$\mbox{colsp}(M)\subseteq\mathbb{F}_{q}^{n}$ denotes the
$\mathbb{F}_{q}$-linear space generated by the columns of $M$ and
$\mbox{rowsp}(M)\subseteq\mathbb{F}_{q}^{m}$ the $\mathbb{F}_{q}$-linear space
generated by the rows of $M$.

######  Definition 1.5.

Let $V\subseteq\mathbb{F}_{q}^{n}$ be a subspace. Then

$\mbox{Mat}(V)=\\{M\in\mathbb{F}_{q}^{\,n\times m}:\mbox{colsp}(M)\subseteq
V\\}\subseteq\mathbb{F}_{q}^{\,n\times m}$

denotes the matrix space supported by the vector space $V$. For
$\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ a rank-metric code, let

$\mathcal{C}(V)=\\{M\in\mathcal{C}:\mbox{colsp}(M)\subseteq
V\\}\subseteq\mathcal{C}$

be the subcode of $\mathcal{C}$ supported on $V$.

Notice that whenever we want to refer to the matrix space supported by a
certain row space we will consider the transposed version of the support given
in Definition 1.5.

An important role in the motivation of this paper is taken by deriving bounds
on rank-metric codes. In the sequel we give two of the most relevant
inequalities, one relating the minimum distance and one relating the maximum
rank to the dimension of a rank-metric code.

######  Theorem 1.6.

(Singleton bound) Let $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ be a
rank-metric code. Then

$\dim(\mathcal{C})\leq m(n-\mbox{d}(\mathcal{C})+1).$ (1)

The Singleton-like bound for rank-metric codes was presented by Delsarte in
[5]. It is the rank-metric version of the well-known Singleton bound in the
Hamming metric. Codes meeting bound (1) are known as Maximum Rank Distance
Codes (MRD codes) and have been extensively studied.

We shall present now an upper bound on the dimension involving the maximum
rank, instead of the minimum distance as seen in (1). A classical theorem by
Flanders in [7] states that the dimension of a linear space of matrices whose
rank is less than or equal to a given $r\leq n$ is upper bounded by $mr$. The
results in [7] are proved under the assumption that the cardinality of the
base field is strictly greater than $r$ and that the characteristic differs
from 2. Atkinson and Lloyd in [1] obtained the same result with the assumption
only on the field size. The square case with $r=n-1$ for an arbitrary field
size was proved by Dieudonné in [6]. Finally, Meshulam in [12] showed that the
assumptions on the rank and on the field size are unnecessary for deriving the
bound on the dimension. In fact, the next bound was proved in [12] and goes
under the name of Anticode bound.

######  Theorem 1.7.

(Anticode bound) Let $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ be a
rank-metric code. Then

$\dim(\mathcal{C})\leq m\,\mbox{maxrk}(\mathcal{C}).$ (2)

######  Definition 1.8.

An optimal anticode $\mathcal{A}\subseteq\mathbb{F}_{q}^{\,n\times m}$ is a
rank-metric code which satisfies

$\dim(\mathcal{A})=m\,\mbox{maxrk}(\mathcal{A}).$

The classification of matrix spaces with least possible maximum rank for a
given dimension follows the same history as the derivation of the Anticode
Bound and is presented in the next theorem.

######  Theorem 1.9.

Let $\mathcal{A}\subseteq\mathbb{F}_{q}^{\,n\times m}$ be an optimal anticode
of dimension $mr$ with $r=\mbox{maxrk}(\mathcal{A})$. Then
$\mathcal{A}=\mbox{Mat}(V)$ for some $V\subseteq\mathbb{F}_{q}^{n}$ of
dimension $r$, or $A=\mbox{Mat}(V)^{t}$ and $m=n$.

Some fundamental properties of optimal anticodes are stated in [14], for
instance that $\mathcal{C}$ is an optimal anticode if and only if
$\mathcal{C}^{\perp}$ is an optimal anticode.

In the end we want to introduce $q$-polymatroids, which are the $q$-analogue
of polymatroids.

######  Definition 1.10 ([9], Definition 4.1).

A $q$-polymatroid is a pair $P=(\mathbb{F}_{q}^{n},\rho)$ where $\rho$ is a
function from the set of all subspaces of $\mathbb{F}_{q}^{n}$ to
$\mathbb{R}$, such that

  * (i)

$0\leq\rho(A)\leq\dim(A)$,

  * (ii)

if $A\subseteq B$, then $\rho(A)\leq\rho(B)$,

  * (iii)

$\rho(A+B)+\rho(A\cap B)\leq\rho(A)+\rho(B)$.

for all $A,B\subseteq\mathbb{F}_{q}^{n}$.

######  Notation 1.11.

Let $s,t$ be positive integers. Throughout the paper we let
$u\in\mathbb{F}_{q}^{s}$ denote a row vector of length $s$ with entries in
$\mathbb{F}_{q}$. The zero matrix of size $s\times t$ is denoted by
$0_{s\times t}$ and the matrix with only one nonzero entry equal to one at
position $(i,j)$ of size $n\times m$ is given by $E_{i,j}$. Further,
$f_{i}\in\mathbb{F}_{q}^{m}$ is the row vector with the only non zero entry
equal 1 at position $i$, whereas $e_{i}\in\mathbb{F}_{q}^{n}$ is the column
vector whose only non zero entry is a 1 in position $i$.

##  2 Quasi optimal and dually quasi optimal anticodes

######  Definition 2.1.

A quasi optimal anticode (qOAC) is a rank-metric code
$\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ such that
$m\nmid\dim(\mathcal{C})$ and

$\mbox{maxrk}(\mathcal{C})=\bigg{\lceil}\frac{\dim(\mathcal{C})}{m}\bigg{\rceil}.$

If $\mathcal{C}$ and $\mathcal{C}^{\perp}$ are both qOACs, then $\mathcal{C}$
is a dually qOAC.

######  Notation 2.2.

For a qOAC $\mathcal{C}$, write

$\dim(\mathcal{C})=\alpha m+\rho,\ \ 0<\rho<m,\ \ 0\leq\alpha<n\leq m,\ \
\mbox{maxrk}(\mathcal{C})=\alpha+1.$

If in addition $\mathcal{C}$ is a dually qOAC, write

$\dim(\mathcal{C}^{\perp})=(n-\alpha-1)m+(m-\rho),\ \ 0<m-\rho<m,\
\mbox{maxrk}(\mathcal{C}^{\perp})=n-\alpha.$

It is well-known that the dual of an optimal anticode is an optimal anticode.
However, this is not the case for qOACs. In the next example, we produce
dually qOACs, as well as qOACs which are not dually qOACs.

######  Example 2.3.

Let $m\geq\max\\{2,n\\}$, let $0<\alpha<n\leq m$, $0<\rho<m$, $0\leq k\leq
m-\rho$. Let

$\mathcal{C}_{k}=\left\\{\begin{pmatrix}A&B\\\ u&0_{1\times k}\\\
w&0_{1\times(m-\rho-k)}\\\
0_{(n-\alpha-1)\times(\rho+k)}&0_{(n-\alpha-1)\times(m-\rho-k)}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{(\alpha-1)\times(m-k)},B\in\mathbb{F}_{q}^{(\alpha-1)\times
k},u\in\mathbb{F}_{q}^{m-k},w\in\mathbb{F}_{q}^{\rho+k}\right\\}$

with dual code

$\mathcal{C}^{\perp}_{k}=\left\\{\begin{pmatrix}0_{(\alpha-1)\times(m-k)}&0_{(\alpha-1)\times
k}\\\ 0_{1\times(m-k)}&u\\\ 0_{1\times(\rho+k)}&w\\\ A&B\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{(n-\alpha-1)\times(\rho+k)},B\in\mathbb{F}_{q}^{(n-\alpha-1)\times(m-\rho-k)},u\in\mathbb{F}_{q}^{k},w\in\mathbb{F}_{q}^{m-\rho-k}\right\\}.$

Notice that $\mathcal{C}_{k}\sim\mathcal{C}_{m-\rho-k}$ for all $k$. We have
$\mbox{maxrk}(\mathcal{C}_{0})=\alpha+1$ and
$\mbox{maxrk}(\mathcal{C}^{\perp}_{0})=n-\alpha$, hence
$\mathcal{C}_{0}\sim\mathcal{C}_{m-\rho}$ are dually qOACs. If $\rho\geq m-2$,
then $\mbox{maxrk}(\mathcal{C}_{1})=\alpha+1$ and
$\mbox{maxrk}(\mathcal{C}^{\perp}_{1})=n-\alpha$, hence
$\mathcal{C}_{1}\sim\mathcal{C}_{m-\rho-1}$ are dually qOACs. If $\rho\leq
m-3$, then $\mbox{maxrk}(\mathcal{C}_{1})=\alpha+1$ and
$\mbox{maxrk}(\mathcal{C}_{1}^{\perp})=n-\alpha+1$. Hence
$\mathcal{C}_{1}\sim\mathcal{C}_{m-\rho-1}$ are qOACs, but not dually qOACs.
For $k\neq 0,1,m-\rho-1,m-\rho$, one has
$\mbox{maxrk}(\mathcal{C}_{k})=\alpha+1$ and
$\mbox{maxrk}(\mathcal{C}^{\perp}_{k})=n-\alpha+1$. Therefore
$\mathcal{C}_{k}\sim\mathcal{C}_{m-\rho-k}$ are qOACs, but not dually qOACs.

The next proposition relates the maximum rank of a code with that of its dual.
It also provides us with a simple characterization of dually qOACs.

######  Proposition 2.4.

Let $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ be a rank-metric code.
Then

$\mbox{maxrk}(\mathcal{C})+\mbox{maxrk}(\mathcal{C}^{\perp})\geq n.$

In addition:

  1. (a)

$\mathcal{C}$ is an optimal anticode if and only if
$\mbox{maxrk}(\mathcal{C})+\mbox{maxrk}(\mathcal{C}^{\perp})=n$.

  2. (b)

$\mathcal{C}$ is a dually qOAC if and only if
$\mbox{maxrk}(\mathcal{C})+\mbox{maxrk}(\mathcal{C}^{\perp})=n+1$.

###### Proof.

The Anticode Bound on $\mathcal{C}$ and $\mathcal{C}^{\perp}$ yields

$mn=\dim(\mathcal{C})+\dim(\mathcal{C}^{\perp})\leq
m(\mbox{maxrk}(\mathcal{C})+\mbox{maxrk}(\mathcal{C}^{\perp})).$ (3)

(a) $\mathcal{C}$ is an optimal anticode if and only if $\mathcal{C}^{\perp}$
is an optimal anticode. Hence, if $\mathcal{C}$ is an optimal anticode, then
equality holds in (3). Conversely, if equality holds in (3), then
$\mathcal{C}$ and $\mathcal{C}^{\perp}$ meet the Anticode Bound, hence they
are optimal anticodes.

(b) If $\mathcal{C}$ is a dually qOAC, then
$\mbox{maxrk}(\mathcal{C})+\mbox{maxrk}(\mathcal{C}^{\perp})=n+1$ by direct
computation. To prove the converse, first observe that, if
$\mbox{maxrk}(\mathcal{C})+\mbox{maxrk}(\mathcal{C}^{\perp})=n+1$, then
$\mathcal{C}$ and $\mathcal{C}^{\perp}$ are not optimal anticodes by part (a).
If $m\mid\dim(\mathcal{C})$, then

$\mbox{maxrk}(\mathcal{C})+\mbox{maxrk}(\mathcal{C}^{\perp})\geq\frac{\dim(\mathcal{C})}{m}+1+\frac{\dim(\mathcal{C}^{\perp})}{m}+1=n+2,$

since $\mathcal{C}$ and $\mathcal{C}^{\perp}$ are not optimal anticodes. If
instead $m\nmid\dim(\mathcal{C})$, then

$n+1=\mbox{maxrk}(\mathcal{C})+\mbox{maxrk}(\mathcal{C}^{\perp})\geq\left\lceil\frac{\dim(\mathcal{C})}{m}\right\rceil+\left\lceil\frac{\dim(\mathcal{C}^{\perp})}{m}\right\rceil\geq\alpha+1+n-\alpha,$
(4)

where $\dim(\mathcal{C})=\alpha m+\rho$,
$\dim(\mathcal{C}^{\perp})=(n-\alpha-1)m+(m-\rho)$, and $\rho>0$. Therefore
the inequalities in (4) are equalities, which completes the proof. ∎

A first classification of large matrix spaces of bounded rank appears in [1],
where Atkinson and Lloyd study linear spaces of dimension close to $mr$ over
fields of large cardinality. Their classification was extended to all fields
and matrices of arbitrary size by de Seguins Pazzis in [3]. As a direct
consequence of the results by de Seguins Pazzis, we can characterize the qOACs
whose dimension is at least $\alpha(m-1)+n$.

######  Theorem 2.5 ([3], Theorem 4, Theorem 5, and Theorem 6).

Let $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ be a qOAC of
$\dim(\mathcal{C})=\alpha m+\rho$, $0<\rho<m$.

  1. (a)

If $\rho>n-\alpha$, then $\mathcal{C}$ is equivalent to a linear subspace of
$\mbox{Mat}(\langle e_{1},\ldots,e_{\alpha+1}\rangle)$.

  2. (b)

If $\rho=n-\alpha$, then one of the following holds:

     * (i)

$\mathcal{C}$ is equivalent to a linear subspace of $\mbox{Mat}(\langle
e_{1},\ldots,e_{\alpha+1}\rangle)$,

     * (ii)

$\mathcal{C}\sim\mbox{Mat}(\langle
e_{1},\ldots,e_{\alpha}\rangle)+\mbox{Mat}(\langle e_{1}\rangle)^{t}$,

     * (iii)

$m=n+1$ and $\mathcal{C}\sim\mbox{Mat}(\langle
e_{1},\ldots,e_{\alpha+1}\rangle)^{t}$,

     * (iv)

$m=n=3$, $q=2$, and $\mathcal{C}\sim\left\\{\begin{pmatrix}a&0&0\\\ c&b&0\\\
d&e&a+b\end{pmatrix}:(a,b,c,d,e)\in\mathbb{F}_{2}^{5}\right\\}$.

Using Theorem 2.5, we can classify dually qOAs.

######  Theorem 2.6.

Let $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ be a dually qOAC with
$\dim(\mathcal{C})=\alpha m+\rho$, $0<\rho<m$. One of the following holds:

  1. (a)

$\mathcal{C}\sim\left\\{\begin{pmatrix}A&B\\\ u&0_{1\times(m-\rho)}\\\
0_{(n-\alpha-1)\times\rho}&0_{(n-\alpha-1)\times(m-\rho)}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{\alpha\times\rho},B\in\mathbb{F}_{q}^{\alpha\times(m-\rho)},u\in\mathbb{F}_{q}^{1\times\rho}\right\\},$

  2. (b)

$\rho\leq n-\alpha$ and

$\mathcal{C}\sim\left\\{\begin{pmatrix}u&A\\\ v&0_{\rho\times(m-1)}\\\
0_{(n-\alpha-\rho)\times 1}&0_{(n-\alpha-\rho)\times(m-1)}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{\alpha\times(m-1)},\\\
u\in\mathbb{F}_{q}^{\alpha\times 1},v\in\mathbb{F}_{q}^{\rho\times
1}\right\\},$

  3. (c)

$\rho\geq m-\alpha-1$ and

$\mathcal{C}\sim\left\\{\begin{pmatrix}A&u\\\ B&0_{(m-\rho)\times 1}\\\
0_{(n-\alpha-1)\times(m-1)}&0_{(n-\alpha-1)\times 1}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{(\alpha+\rho+1-m)\times(m-1)},B\in\mathbb{F}_{q}^{(m-\rho)\times(m-1)},u\in\mathbb{F}_{q}^{(\alpha+\rho+1-m)\times
1}\right\\}.$

  4. (d)

$m=n+1$, $\rho=n-\alpha$, and

$\mathcal{C}\sim\left\\{\begin{pmatrix}A&0_{n\times(n-\alpha)}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{n\times(\alpha+1)}\right\\}.$

###### Proof.

It is easy to check that the codes in the statement of the theorem are dually
qOACs. We now prove that, up to equivalence, they are the only ones. We start
by analyzing the case when $\mathcal{C}\supseteq\mbox{Mat}(U)$, for some
$U\subseteq\mathbb{F}_{q}^{n}$ of $\dim(U)=\alpha$. Up to equivalence, we may
assume that $U=\langle e_{1},\ldots,e_{\alpha}\rangle$. Write
$\mathcal{C}=\mbox{Mat}(\langle e_{1},\ldots,e_{\alpha}\rangle)+\langle
M_{1},\ldots,M_{\rho}\rangle$, where
$M_{1},\ldots,M_{\rho}\in\mbox{Mat}(\langle
e_{1},\ldots,e_{\alpha}\rangle)^{\perp}=\mbox{Mat}(\langle
e_{\alpha+1},\ldots,e_{n}\rangle)$ are linearly independent. Since
$\mathcal{C}$ is a qOAC, then $\mbox{maxrk}(\mathcal{C})=\alpha+1$. We claim
that $\mbox{maxrk}(\langle M_{1},\ldots,M_{\rho}\rangle)=1$. In fact, any
$M\in\langle M_{1},\ldots,M_{\rho}\rangle\subseteq\mbox{Mat}(\langle
e_{\alpha+1},\ldots,e_{n}\rangle)$ has

$\dim(\mbox{rowsp}(M))=\mbox{rank}(M)\leq\mbox{maxrk}(\mbox{Mat}(\langle
e_{\alpha+1},\ldots,e_{n}\rangle))=n-\alpha.$

Let $L\in\mbox{Mat}(\langle e_{1},\ldots,e_{\alpha}\rangle)$ be a matrix whose
first $\alpha$ rows are linearly independent vectors in a vector space $V$
such that $V\oplus\mbox{rowsp}(M)=\mathbb{F}_{q}^{m}$. Notice that one can
always find such an $L$, since $\dim(V)=m-\dim(\mbox{rowsp}(M))\geq
m-(n-\alpha)\geq\alpha$. Then $\mbox{rowsp}(L)\cap\mbox{rowsp}(M)=0$, so
$L+M\in\mathcal{C}$ has

$\mbox{rank}(L+M)=\mbox{rank}(M)+\mbox{rank}(L)\leq\mbox{maxrk}(\mathcal{C})=\alpha+1,$

which proves that $\mbox{rank}(M)\leq 1$, since $\mbox{rank}(L)=\alpha$. Since
$\mbox{maxrk}(\langle M_{1},\ldots,M_{\rho}\rangle)=1$, then either $\langle
M_{1},\ldots,M_{\rho}\rangle\subseteq\mbox{Mat}(w)$ for some $w\in\langle
e_{\alpha+1},\ldots,e_{n}\rangle\subseteq\mathbb{F}_{q}^{n}$, or $\langle
M_{1},\ldots,M_{\rho}\rangle\subseteq\mbox{Mat}(w)^{t}$ for some
$w\in\mathbb{F}_{q}^{m}$. Since $\langle
M_{1},\ldots,M_{\rho}\rangle\subseteq\mbox{Mat}(\langle
e_{\alpha+1},\ldots,e_{n}\rangle)$, the latter is only possible if

$\rho=\dim(\langle M_{1},\ldots,M_{\rho}\rangle)\leq\dim(\mbox{Mat}(\langle
e_{\alpha+1},\ldots,e_{n}\rangle)\cap\mbox{Mat}(w)^{t})=n-\alpha.$

If $\langle M_{1},\ldots,M_{\rho}\rangle\subseteq\mbox{Mat}(w)$, then, after
suitable invertible operations involving the last $n-\alpha$ rows, we may
suppose that $w=e_{\alpha+1}$. Moreover, after suitable invertible column
operations

$M_{i}=\begin{pmatrix}0_{n\times(i-1)}&e_{\alpha+1}&0_{n\times(m-i)}\\\
\end{pmatrix}\quad\mbox{ for }1\leq i\leq\rho.$

Since both types of operations fix $\mbox{Mat}(\langle
e_{1},\ldots,e_{\alpha}\rangle)$, we have shown that

$\mathcal{C}\sim\left\\{\begin{pmatrix}A&B\\\ u&0_{1\times(m-\rho)}\\\
0_{(n-\alpha-1)\times\rho}&0_{(n-\alpha-1)\times(m-\rho)}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{\alpha\times\rho},B\in\mathbb{F}_{q}^{\alpha\times(m-\rho)},u\in\mathbb{F}_{q}^{1\times\rho}\right\\}.$
(5)

This yields the codes in part (a) of the statement.

If $\rho\leq n-\alpha$ and $\langle
M_{1},\ldots,M_{\rho}\rangle\subseteq\mbox{Mat}(w)^{t}$, then, after suitable
invertible operations involving the last $n-\alpha$ rows, we may suppose that
$M_{i}$ is the matrix whose rows are all zero, except for row $\alpha+i$ which
is equal to $w$. Up to invertible column operations, we may further suppose
that $w=e_{1}\in\mathbb{F}_{q}^{m}$. Since both types of operations fix
$\mbox{Mat}(\langle e_{1},\ldots,e_{\alpha}\rangle)$, we have that

$\mathcal{C}\sim\left\\{\begin{pmatrix}u&A\\\ v&0_{\rho\times(m-1)}\\\
0_{(n-\alpha-\rho)\times 1}&0_{(n-\alpha-\rho)\times(m-1)}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{\alpha\times(m-1)},\\\
u\in\mathbb{F}_{q}^{\alpha\times 1},v\in\mathbb{F}_{q}^{\rho\times
1}\right\\}.$

This yields the codes in part (b) of the statement.

Suppose now that $\mathcal{C}\subseteq\mbox{Mat}(U)$ for some
$U\subseteq\mathbb{F}_{q}^{n}$ of $\dim(U)=\alpha+1$. Up to equivalence, we
may assume that $U=\mbox{Mat}(\langle e_{n-\alpha},\ldots,e_{n}\rangle)$, that
is, $\mathcal{C}^{\perp}\supseteq\mbox{Mat}(\langle
e_{1},\ldots,e_{n-\alpha-1}\rangle)$. Then the above argument shows that
either

$\mathcal{C}^{\perp}\sim\left\\{\begin{pmatrix}A&B\\\ u&0_{1\times\rho}\\\
0_{\alpha\times(m-\rho)}&0_{\alpha\times\rho}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{(n-\alpha-1)\times(m-\rho)},B\in\mathbb{F}_{q}^{(n-\alpha-1)\times\rho},u\in\mathbb{F}_{q}^{1\times(m-\rho)}\right\\}$

or

$\mathcal{C}^{\perp}\sim\left\\{\begin{pmatrix}u&A\\\
v&0_{(m-\rho)\times(m-1)}\\\ 0_{(\alpha+\rho+1-m)\times
1}&0_{(\alpha+\rho+1-m)\times(m-1)}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{(n-\alpha-1)\times(m-1)},\\\
u\in\mathbb{F}_{q}^{(n-\alpha-1)\times 1},v\in\mathbb{F}_{q}^{(m-\rho)\times
1}\right\\}.$

Moreover, the latter is only possible when $\rho\geq m-\alpha-1$. Taking
duals, we obtain that either $\mathcal{C}$ has the form (5), or

$\mathcal{C}\sim\left\\{\begin{pmatrix}A&u\\\ B&0_{(m-\rho)\times 1}\\\
0_{(n-\alpha-1)\times(m-1)}&0_{(n-\alpha-1)\times 1}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{(\alpha+\rho+1-m)\times(m-1)},\\\
B\in\mathbb{F}_{q}^{(m-\rho)\times(m-1)},u\in\mathbb{F}_{q}^{(\alpha+\rho+1-m)\times
1}\right\\}.$

This yields the codes in part (c) of the statement.

Finally, suppose that $\mathcal{C}\not\supseteq\mbox{Mat}(U)$ for any
$U\subseteq\mathbb{F}_{q}^{n}$ of $\dim(U)=\alpha$ and
$\mathcal{C}\not\subseteq\mbox{Mat}(U)$ for any $U\subseteq\mathbb{F}_{q}^{n}$
of $\dim(U)=\alpha+1$. This implies also that
$\mathcal{C}^{\perp}\not\subseteq\mbox{Mat}(U)$ for any
$U\subseteq\mathbb{F}_{q}^{n}$ of $\dim(U)=n-\alpha$. Since $n\leq m$, then
either $\rho\leq m-\alpha-1$ or $\rho\geq n-\alpha$. This implies that, for
any $0<\rho<m$, Theorem 2.5 applies to either $\mathcal{C}$ or
$\mathcal{C}^{\perp}$. Because of our assumptions, and since the code of
Theorem 2.5 (b) (iv) is not a dually qOAC, $\mathcal{C}$ or
$\mathcal{C}^{\perp}$ is a code as in Theorem 2.5 (b) (iii). Therefore $m=n+1$
and $\rho=m-\alpha-1=n-\alpha$ and we obtain the codes in part (d) of the
statement. ∎

Theorem 2.5 and Theorem 2.6 provide us with many examples of codes which are
qOACs but not dually qOACs. In fact, any code as in Theorem 2.5 which is not
one of the codes in Theorem 2.6 is of this kind. We now give some concrete
examples of qOACs which are not dually qOACs. These examples are covered by
Theorem 2.5, unless $m=n$ and $\rho=n-\alpha-1$.

######  Example 2.7.

Let $1\leq\alpha\leq n-1$ and $m-\alpha-1\leq\rho\leq m-2$. Let

$\begin{array}[]{ccl}\mathcal{C}&=&\Bigg{\\{}\begin{pmatrix}\mathcal{V}&B\\\
A&C\\\ 0_{(n-\alpha-1)\times(m-\rho)}&0_{(n-\alpha-1)\times\rho}\\\
\end{pmatrix}:\mathcal{V}\subseteq\mathbb{F}_{q}^{(m-\rho)\times(m-\rho)}\mbox{
the set of matrices}\\\ &&\\\ &&\mbox{ with 0 on the
diagonal},A\in\mathbb{F}_{q}^{(\alpha+1-m+\rho)\times(m-\rho)},B\in\mathbb{F}_{q}^{(m-\rho)\times\rho},C\in\mathbb{F}_{q}^{(\alpha+1-m+\rho)\times\rho}\Bigg{\\}}.\end{array}$

Then $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ has dimension $\alpha
m+\rho$ and maximum rank $\alpha+1$, hence it is a qOAC. Its dual is given by

$\begin{array}[]{ccc}\mathcal{C}^{\perp}&=&\Bigg{\\{}\begin{pmatrix}\mathcal{D}&0_{(m-\rho)\times\rho}\\\
0_{(\alpha+1-m+\rho)\times(m-\rho)}&0_{(\alpha+1-m+\rho)\times\rho}\\\ A&B\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{(n-\alpha-1)\times(m-\rho)},B\in\mathbb{F}_{q}^{(n-\alpha-1)\times\rho},\\\
&&\\\ &&\mathcal{D}\subseteq\mathbb{F}_{q}^{(m-\rho)\times(m-\rho)}\mbox{ the
set of diagonal matrices}\Bigg{\\}}.\end{array}$

$\mathcal{C}^{\perp}$ has dimension $(n-\alpha)m-\rho$ and maximum rank
$m+n-\rho-\alpha-1\geq n-\alpha+1$. Hence $\mathcal{C}^{\perp}$ is not a qOAC,
that is, $\mathcal{C}$ is not a dually qOAC.

Another result from [4] provides us with information on qOACs beyond those
treated in Theorem 2.5. Below we state the result in our language.

######  Theorem 2.8 ([4], Theorem 1.7).

Let $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ be a qOAC with
$\dim(\mathcal{C})=\alpha m+\rho$, $0<\rho<m$. If $q=2$ suppose that $\rho\geq
2(n-\alpha+1)-m$, else suppose that $\rho\geq 2(n-\alpha)-m$. Then
$\mathcal{C}$ is equivalent to a linear subspace of $\mbox{Mat}(\langle
e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle
e_{1},\ldots,e_{\alpha+1-s}\rangle)^{t}$, for some
$s\in\\{0,\ldots,\alpha+1\\}$.

If $\mathcal{C}$ is a qOAC of dimension $\dim(\mathcal{C})=\alpha m+\rho$ and
$\mbox{maxrk}(\mathcal{C})=\alpha+1$, then $\mathcal{C}$ cannot be equivalent
to a subspace of $\mbox{Mat}(\langle
e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle e_{1},\ldots,e_{k}\rangle)^{t}$
for any $s,k\geq 0$ with $s+k\leq\alpha$. Lemma 1 in [7] shows that, up to
equivalence, $\mathcal{C}$ is contained in $\mbox{Mat}(\langle
e_{1},\ldots,e_{\alpha+1}\rangle)+\mbox{Mat}(\langle
e_{1},\ldots,e_{\alpha+1}\rangle)^{t}$. Theorem 2.5 and Theorem 2.8 prove
that, under some assumptions on $\rho$, $\mathcal{C}$ is equivalent to a
subspace of $\mbox{Mat}(\langle e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle
e_{1},\ldots,e_{\alpha+1-s}\rangle)^{t}$. In the next example, we exhibit a
code which does not satisfy the assumptions of Theorem 2.5 or Theorem 2.8 and
which is not equivalent to a subspace of $\mbox{Mat}(\langle
e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle
e_{1},\ldots,e_{\alpha+1-s}\rangle)^{t}$ for any $0\leq s\leq\alpha+1$. The
example is based on [12, Remark on pg. 228].

######  Example 2.9.

Let $\mathcal{C}\subseteq\mathbb{F}_{2}^{4\times 4}$ be a linear code given by

$\mathcal{C}=\left\\{\begin{pmatrix}a_{1}&a_{4}&a_{5}&a_{6}\\\
0&a_{2}&a_{7}&a_{8}\\\ 0&0&a_{2}+a_{3}&a_{9}\\\ 0&0&0&a_{3}\\\
\end{pmatrix}:a_{i}\in\mathbb{F}_{2}\mbox{ for }1\leq i\leq 9\right\\}.$

Then $\mbox{maxrk}(\mathcal{C})=3$ and $\dim(\mathcal{C})=9$, hence
$\mathcal{C}$ is a qOAC. We claim that $\mathcal{C}$ is not equivalent to a
subspace of $\mbox{Mat}(\langle e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle
e_{1},\ldots,e_{3-s}\rangle)^{t}$ for any $0\leq s\leq 3$. To see this,
consider the following two elements of $\mathcal{C}$:

$M_{1}=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\ 0&0&0&0\\\
\end{pmatrix}\mbox{ and }M_{2}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&1&0\\\
0&0&0&1\\\ \end{pmatrix}.$

We prove that if $M_{1}$ is contained in $\mbox{Mat}(U)+\mbox{Mat}(V)^{t}$ for
some $U,V\subseteq\mathbb{F}_{2}^{4}$ of $\dim(V)+\dim(U)=3$, then
$U,V\subseteq\langle e_{1},e_{2},e_{3}\rangle$. Notice that this establishes
the claim, since $M_{2}$ is not contained in such a space.

The only non trivial case, up to transposition, is $\dim(U)=2$ and
$\dim(V)=1$. Let
$U=\langle(x_{1},y_{1},z_{1},0),(x_{2},y_{2},z_{2},w)\rangle\subseteq\mathbb{F}_{2}^{4}$
and $V=\langle(a,b,c,d)\rangle\subseteq\mathbb{F}_{2}^{4}$. By assumption
there are
$A\in\mbox{Mat}(\langle(x_{1},y_{1},z_{1},0),(x_{2},y_{2},z_{2},w)\rangle)$
and $B\in\mbox{Mat}(\langle(a,b,c,d)\rangle)^{t}$, such that $M_{1}=A+B$.
Equating the last column of $A+B$ to that of $M_{1}$ yields the following
homogeneous linear system:

$\begin{array}[]{ccccccc}0&=&\alpha d&+&t_{1}x_{1}&+&t_{2}x_{2}\\\ 0&=&\beta
d&+&t_{1}y_{1}&+&t_{2}y_{2}\\\ 0&=&\gamma d&+&t_{1}z_{1}&+&t_{2}z_{2}\\\
0&=&\delta d&+&0&+&t_{2}w.\\\ \end{array}$ (6)

Notice that $(\alpha,\beta,\gamma,\delta)\notin U$, otherwise $M_{1}$ would be
contained in $\mbox{Mat}(U)$, but this is not possible since
$\mbox{maxrk}(\mbox{Mat}(U))=2$. Therefore, the only solution of (6) regarded
as a system in the variables $d,t_{1},t_{2}$ is $d=t_{1}=t_{2}=0$.

Applying the same argument to the last row of $M_{1}$ yields another
homogeneous linear system, which, regarded as system in the variables
$\delta,w$ only has the trivial solution $\delta=w=0$. In particular, we have
proved that $d=w=0$, i.e., $U,V\subseteq\langle e_{1},e_{2},e_{3}\rangle.$

In the sequel, we concentrate on linear spaces of the form $\mbox{Mat}(\langle
e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle e_{1},\ldots,e_{k}\rangle)^{t}$,
as well as some of their linear subspaces.

######  Definition 2.10.

Let $s,h,k\geq 0$ be integers such that $k\leq m$ and $0<s+h\leq n$. Let

$\mathcal{C}_{s,h,k}=\left\\{\begin{pmatrix}A&B\\\ C&0_{h\times(m-k)}\\\
0_{(n-s-h)\times k}&0_{(n-s-h)\times(m-k)}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{s\times
k},B\in\mathbb{F}_{q}^{s\times(m-k)},C\in\mathbb{F}_{q}^{h\times
k},\right\\}\subseteq\mathbb{F}_{q}^{\,n\times m}.$

######  Remark 2.11.

Up to equivalence, all the dually qOAC are of the form $\mathcal{C}_{s,h,k}$
for some $s,h,k$ by Theorem 2.6. Moreover, one can check that $C_{s,h,k}$ is a
dually qOAC if and only if there are parameters $0\leq\alpha<n$ and $0<\rho<m$
such that $(s,h,k)$ is among $(\alpha,\rho,1),(\alpha,1,\rho),(0,n,\alpha+1)$
if $m=n+1$, or $(\alpha+\rho+1-m,m-\rho,m-1)$ if $\rho\geq m-\alpha-1$.

In the rest of the paper, we compute the invariants of the codes from
Definition 2.10. In the next proposition, we characterize which codes of the
form $\mathcal{C}_{s,h,k}$ are qOACs. Together with Theorem 2.6, Proposition
2.12 yields examples of qOACs which are not dually qOACs, beyond those of
Example 2.7. Some examples of this kind are given in Example 2.13.

######  Proposition 2.12.

Let $s,h,k$ be non-negative integers such that $0<s+h\leq n$ and $k<m$. Then
$\mathcal{C}_{s,h,k}$ is a qOAC if and only if

$0<\min\\{h,k\\}\leq\left\lfloor\frac{m-1}{m-\max\\{h,k\\}}\right\rfloor.$

###### Proof.

The code $\mathcal{C}$ has dimension $\dim(\mathcal{C})=sm+hk$ and maximum
rank $s+\min\\{h,k\\}$. Then $\mathcal{C}$ is a qOAC iff $m\nmid hk$ and

$\left\lceil\frac{sm+hk}{m}\right\rceil=s+\min\\{h,k\\}\quad\Longleftrightarrow\quad\left\lceil\frac{hk}{m}\right\rceil=\min\\{h,k\\}.$

Therefore, $\mathcal{C}$ is a qOAC iff $\min\\{h,k\\}>0$ and
$(\min\\{h,k\\}-1)m+1\leq hk\leq\min\\{h,k\\}m-1$. A computation shows that
$\mathcal{C}$ is a qOAC iff

$0<\min\\{h,k\\}\leq\left\lfloor\frac{m-1}{m-\max\\{h,k\\}}\right\rfloor.$

Notice that $\max\\{h,k\\}<m$, since $m\nmid hk$. ∎

######  Example 2.13.

Let $0<k<m$ and let

$\mathcal{C}=\left\\{\begin{pmatrix}A&0_{n\times(m-k)}\\\
\end{pmatrix}:A\in\mathbb{F}_{q}^{n\times
k},\right\\}\subseteq\mathbb{F}_{q}^{\,n\times m}.$

By Theorem 2.6, $\mathcal{C}$ is a dually qOAC if and only if $m=n+1$. By
Proposition 2.12, $\mathcal{C}$ is a qOAC if and only if

$k\leq\min\left\\{\left\lfloor\frac{m-1}{m-n}\right\rfloor,n\right\\}\ \mbox{
or }\ n\leq\min\left\\{\left\lfloor\frac{m-1}{m-k}\right\rfloor,k\right\\}.$

In the following sections, we compute the invariants of the codes from
Definition 2.10. Since this does not affect the computation of the invariants,
we always assume that $n=s+h$. This amounts to considering the codes
$C_{s,k,n-s}=\mbox{Mat}(\langle e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle
e_{1},\ldots,e_{k}\rangle)^{t}$.

##  3 Generalized weights

In this section we compute the generalized weights of codes of the form
$\mbox{Mat}(\langle e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle
e_{1},\ldots,e_{k}\rangle)^{t}$. For what discussed in the previous section,
this determines the weights of all dually qOACs and of certain qOACs. We start
by recalling the definition of generalized weights.

######  Definition 3.1.

([13], Definition 23) Let $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$
be a rank-metric code. For $1\leq i\leq\dim(\mathcal{C})$, the $i$th
generalized weight of $\mathcal{C}$ is

$a_{i}(\mathcal{C})=\displaystyle\frac{1}{m}\min\\{\dim(\mathcal{A})\,:\,\mathcal{A}\subseteq\mathbb{F}_{q}^{\,n\times
m}\mbox{ is an optimal anticode with }\dim(\mathcal{A}\cap\mathcal{C})\geq
i\\}.$

Generalized weights were defined in [13], where some of their basic properties
were also established. A different - but related - definition of generalized
weights was given in [11].

We start with a simple result, which will be used in the proof of Theorem 3.3.

######  Proposition 3.2.

Let $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ be a rank-metric code
which contains an optimal anticode of dimension $sm$ for some $1\leq s\leq n$.
Then

$a_{(i-1)m+1}(\mathcal{C})=\cdots=a_{im}(\mathcal{C})=i\mbox{ for }1\leq i\leq
s.$

###### Proof.

Fix $0\leq i\leq s$ and let $\mathcal{A}$ be an optimal anticode of dimension
$\dim(\mathcal{A})=im$ such that $\mathcal{A}\subseteq\mathcal{C}$. Then
$a_{im}(\mathcal{C})=i$, since $a_{im}(\mathcal{C})\geq i$ by [13, Theorem 30]
and $a_{im}(\mathcal{C})\leq i$ since $\mathcal{C}\supseteq\mathcal{A}$. Using
the properties of generalized weights from [13, Theorem 30], one easily
obtains that

$a_{(i-1)m+1}(\mathcal{C})=\cdots=a_{im}(\mathcal{C})=i.$

∎

We are now ready to state the main theorem of this section. The next theorem
computes the generalized weights of all dually qOACs and some qOACs.

######  Theorem 3.3.

Let $\mathcal{C}=\mbox{Mat}(\langle
e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle
e_{1},\ldots,e_{k}\rangle)^{t}\subseteq\mathbb{F}_{q}^{\,n\times m}$. Then

$\begin{array}[]{clcll}a_{(i-1)m+1}(\mathcal{C})&=\cdots=&a_{im}(\mathcal{C})&=i&\mbox{
for }1\leq i\leq s,\\\
a_{sm+(i-1)k+1}(\mathcal{C})&=\cdots=&a_{sm+ik}(\mathcal{C})&=s+i&\mbox{ for
}1\leq i\leq n-s.\end{array}$

###### Proof.

Since $\mathcal{C}\supseteq\mbox{Mat}(\langle e_{1},\ldots,e_{s}\rangle)$,
then by Proposition 3.2

$a_{(i-1)m+1}(\mathcal{C})=\cdots=a_{im}(\mathcal{C})=i\quad\mbox{ for }1\leq
i\leq s.$

Fix $1\leq i\leq n-s$ and let
$\mathcal{A}=\mbox{Mat}(V)\subseteq\mathbb{F}_{q}^{\,n\times m}$ be an optimal
anticode with $V\subseteq\mathbb{F}_{q}^{n}$ of $\dim(V)=s+i$. We claim that
$\dim(\mathcal{C}\cap\mathcal{A})\leq sm+ik$. Since $\dim(V)=s+i$, then there
exist $i$ linearly independent vectors $v_{1},\ldots,v_{i}\in
V\setminus\langle e_{1},\ldots,e_{s}\rangle$. Consider the matrices

$M_{j}^{1}=\begin{pmatrix}0_{n\times k}&v_{j}&0_{n\times m-k-1}\\\
\end{pmatrix},\ M_{j}^{2}=\begin{pmatrix}0_{n\times k+1}&v_{j}&0_{n\times
m-k-2}\\\ \end{pmatrix},\ \ldots\ ,M_{j}^{m-k}=\begin{pmatrix}0_{n\times
m-1}&v_{j}\\\ \end{pmatrix}$

for $1\leq j\leq i$. Clearly $\langle
M_{1}^{1},\ldots,M_{i}^{1},\cdots,M_{1}^{m-k},\ldots,M_{i}^{m-k}\rangle\subseteq\mathcal{A}$.
Moreover

$\mathcal{C}\ \cap\ \langle
M_{1}^{1},\ldots,M_{i}^{1},\cdots,M_{1}^{m-k},\ldots,M_{i}^{m-k}\rangle=0.$
(7)

Hence

$\begin{array}[]{ccl}\dim(\mathcal{C})+\dim(\mathcal{A})-\dim(\mathcal{C}\cap\mathcal{A})&=&\dim(\mathcal{C}+\mathcal{A})\\\
&\geq&\dim(\mathcal{C})+\dim(\langle
M_{1}^{1},\ldots,M_{i}^{1},\cdots,M_{1}^{m-k},\ldots,M_{i}^{m-k}\rangle),\\\
\end{array}$

which proves our claim that

$\dim(\mathcal{C}\cap\mathcal{A})\leq sm+ik.$

It follows directly that

$a_{sm+ik+1}(\mathcal{C})\geq s+i+1\mbox{ for }1\leq i\leq n-s-1.$ (8)

The inequality

$a_{sm+1}(\mathcal{C})\geq s+1$ (9)

follows from [13, Theorem 30].

In order to prove that all the previous inequalities are in fact equalities,
consider $\mathcal{A}=\mbox{Mat}(\langle e_{1},\ldots,e_{s+i}\rangle)$. It is
easy to check that $\dim(\mathcal{C}\cap\mathcal{A})=sm+ik$. Therefore
$a_{sm+ik}(\mathcal{C})\leq s+i$ which, together with (8) and (9) yields

$a_{sm+(i-1)k+1}(\mathcal{C})=\cdots=a_{sm+ik}(\mathcal{C})=s+i+1,$

for $1\leq i\leq n-s$. ∎

##  4 Rank distribution

In this section we compute the rank distribution of codes of the form
$\mbox{Mat}(\langle e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle
e_{1},\ldots,e_{k}\rangle)^{t}$. For what discussed in Section 2, this
determines the rank distribution of all dually qOACs and of certain qOACs. We
start by recalling a basic result from linear algebra.

######  Definition 4.1.

The Gaussian $q$-binomial coefficient is the integer

$\binom{n}{r}_{q}=\frac{(q^{n}-1)(q^{n}-q)\cdots(q^{n}-q^{r-1})}{(q^{r}-1)(q^{r}-q)\cdots(q^{r}-q^{r-1})}.$

######  Lemma 4.2.

The number of rank $r$ matrices in $\mathbb{F}_{q}^{\,n\times m}$ is

$\phi_{q}(n,m,r)=\binom{n}{r}_{q}(q^{m}-1)(q^{m}-q)\cdots(q^{m}-q^{r-1}).$

######  Theorem 4.3.

Let $\mathcal{C}=\mbox{Mat}(\langle
e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle e_{1},\ldots,e_{k}\rangle)^{t}$.
For $0\leq r\leq n$ denote by $A_{r}$ the number of elements of rank $r$ in
$\mathcal{C}$. Then

$A_{r}=\phi_{q}(s,m,r)+\sum_{i=1}^{\min\\{n-s,k,r\\}}\binom{k}{i}_{q}\binom{m-i}{r-i}_{q}\
q^{si}\ \prod_{t=0}^{i-1}(q^{n-s}-q^{t})\prod_{j=0}^{r-i-1}(q^{s}-q^{j}),$

for $1\leq r\leq\min\\{s+k,n\\}$.

###### Proof.

The number of $n\times m$ matrices in $\mathcal{C}$ of rank $1\leq r\leq n$ is
the sum of the number of matrices of rank $r$ in $\mathbb{F}_{q}^{s\times m}$
and the number of $n\times m$ matrices of rank $r$ in
$\mathbb{F}_{q}^{\,n\times m}$ which have at least one nonzero row among the
last $n-s$. We denote the latter set of matrices by $\mathcal{C}_{r}$.

The first number is $\phi_{q}(s,m,r)$ by Lemma 4.2. We now compute the
cardinality of $\mathcal{C}_{r}$. Let $1\leq i\leq\min\\{n-s,k,r\\}$ and let
$V\subseteq\mathbb{F}_{q}^{m}$ of $\dim(V)=i$. Let
$U\subseteq\mathbb{F}_{q}^{m}$ be an $r$-dimensional vector space containing
$V$. Since $r$-dimensional vector spaces $U$ containing a given
$i$-dimensional subspace are in one-to-one correspondence with
$(r-i)$-dimensional vector spaces in $\mathbb{F}_{q}^{m-i}$, the number of
$r$-dimensional vector spaces $U$ such that of $\mathbb{F}_{q}^{m}\supseteq
U\supseteq V$ is

$\binom{m-i}{r-i}_{q}.$ (10)

Next we determine the number of matrices with row space $U$ and such that the
row space of the last $n-s$ rows is $V$. Complete a basis $u_{1},\ldots,u_{i}$
of $V$ to a basis $u_{1},\ldots,u_{r}$ of $U$. We observe that every such
$M\in\mathcal{C}_{r}$ is of the form

$M=D\cdot\begin{pmatrix}u_{1}\\\ \vdots\\\ u_{r}\\\
\end{pmatrix}\in\mathcal{C}_{r},\mbox{ where }D=\begin{pmatrix}A&B\\\
C&0_{(n-s)\times(r-i)}\\\ \end{pmatrix},$

$A\in\mathbb{F}_{q}^{s\times
i},B\in\mathbb{F}_{q}^{s\times(r-i)},C\in\mathbb{F}_{q}^{(n-s)\times i}$.
Moreover $B,C$, and $D$ must have full rank. Since there is a one-to-one
correspondence between $M$ and $D$, the number of matrices in
$\mathcal{C}_{r}$ with row space $U$ and such that the last $n-s$ rows
generate $V$ is

$q^{si}\prod_{t=0}^{i-1}(q^{n-s}-q^{t})\prod_{j=0}^{r-i-1}(q^{s}-q^{j}).$ (11)

The product of (10) and (11) is then the number of matrices in
$\mathcal{C}_{r}$, whose last $n-s$ rows span $V$.

Finally, observe that for a given $1\leq i\leq\min\\{n-s,k,r\\}$ there are

$\binom{k}{i}_{q}$

$i$-dimensional subspaces $V$ in $\mathbb{F}_{q}^{k}$. Hence the cardinality
of $\mathcal{C}_{r}$ is

$\sum_{i=1}^{\min\\{n-s,k,r\\}}\binom{k}{i}_{q}\binom{m-i}{r-i}_{q}\,q^{si}\prod_{t=0}^{i-1}(q^{h}-q^{t})\prod_{j=0}^{r-i-1}(q^{s}-q^{j}).$

∎

##  5 Associated $q$-Polymatroids

$q$-polymatroids are the $q$-analogs of polymatroids. They were introduced
indipendently by Shiromoto in [16] and by Gorla, Jurrius, López Valdez, and
Ravagnani in [9]. In this section we compute the rank functions of the
$q$-polymatroids associated to codes of the form $\mbox{Mat}(\langle
e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle e_{1},\ldots,e_{k}\rangle)^{t}$.
By Lemma 5.4 this determines the $q$-polymatroids associated to all dually
qOACs and to certain qOACs.

We start by recalling the relevant definitions. The fact that the functions
$\rho_{c}$ and $\rho_{r}$ define $q$-polymatroids is shown in [9, Theorem
5.3].

######  Definition 5.1.

Let $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ be a rank-metric code,
and let $J\subseteq\mathbb{F}_{q}^{n}$ and $K\subseteq\mathbb{F}_{q}^{m}$ be
linear subspaces. The $q$-polymatroids associated to $\mathcal{C}$ are
$(\mathbb{F}_{q}^{n},\rho_{c})$ and $(\mathbb{F}_{q}^{m},\rho_{r})$, where

$\rho_{c}(\mathcal{C},J)=\frac{\dim(\mathcal{C})-\dim(\mathcal{C}\cap\mbox{Mat}(J^{\perp}))}{m}$

and

$\rho_{r}(\mathcal{C},K)=\frac{\dim(\mathcal{C})-\dim(\mathcal{C}\cap\mbox{Mat}(K^{\perp})^{t})}{n}.$

The next proposition follows directly from the definition of the rank
functions.

######  Proposition 5.2.

Let $\mathcal{C}\subseteq\mathbb{F}_{q}^{\,n\times m}$ be a rank-metric code
and assume that $m\nmid\dim(\mathcal{C})$. The following are equivalent:

  1. 1.

$\mathcal{C}$ is a qOAC,

  2. 2.

$\lceil\rho_{c}(\mathcal{C},\mathbb{F}_{q}^{n})\rceil=\mbox{maxrk}(\mathcal{C})$,

  3. 3.

$m=n$ and
$\lceil\rho_{r}(\mathcal{C},\mathbb{F}_{q}^{m})\rceil=\mbox{maxrk}(\mathcal{C})$.

The next lemmas will be used in the proof of Theorem 5.5.

######  Lemma 5.3.

Let $h\geq 0$, $0\leq k\leq m$, and $0\leq s\leq m-h$. Let
$\mathcal{C}=\mbox{Mat}(\langle e_{1},\ldots,e_{s}\rangle)+\mbox{Mat}(\langle
e_{1},\ldots,e_{k}\rangle)^{t}\subseteq\mathbb{F}_{q}^{(s+h)\times m}$ be a
rank-metric code. Let $J\subseteq\mathbb{F}_{q}^{s+h}$ and
$K\subseteq\mathbb{F}_{q}^{m}$. Then:

  * (a)

If $J\cap\langle e_{1},\ldots,e_{s}\rangle=0$, then
$\dim(\mathcal{C}\cap\mbox{Mat}(J))=k\cdot\dim(J).$

  * (b)

If $K\cap\langle e_{1},\ldots,e_{k}\rangle=0$, then
$\dim(\mathcal{C}\cap\mbox{Mat}(K)^{t})=s\cdot\dim(K).$

###### Proof.

We only prove the first statement, as the second is proved similarly. Let
$M\in\mathcal{C}\cap\mbox{Mat}(J)$. The last $m-k$ columns of $M$ belong to
$J\cap\langle e_{1},\ldots,e_{s}\rangle$, hence they are zero. Therefore
$\mathcal{C}\cap\mbox{Mat}(J)\subseteq\mbox{Mat}(\langle
e_{1},\ldots,e_{k}\rangle)^{t}\cap\mbox{Mat}(J)\subseteq\mathcal{C}\cap\mbox{Mat}(J),$
where the second inclusion follows from
$\mathcal{C}\supseteq\mbox{Mat}(\langle e_{1},\ldots,e_{k}\rangle)^{t}$. It
follows that

$\mathcal{C}\cap\mbox{Mat}(J)=\mbox{Mat}(\langle
e_{1},\ldots,e_{k}\rangle)^{t}\cap\mbox{Mat}(J),$

in particular $\dim(\mathcal{C}\cap\mbox{Mat}(J))=k\cdot\dim(J)$. ∎

The proof of the next lemma is immediate.

######  Lemma 5.4.

Let $V\subseteq\mathbb{F}_{q}^{n}$ and let
$\mathcal{C}\subseteq\mbox{Mat}(V)\subseteq\mathbb{F}_{q}^{\,n\times m}$ be a
rank-metric code. Then

$\mathcal{C}\cap\mbox{Mat}(J)=\mathcal{C}\cap\mbox{Mat}(J\cap V)$

for any $J\subseteq\mathbb{F}_{q}^{n}$. In particular, if
$\mathcal{C}\subseteq\mbox{Mat}(\langle e_{1},\ldots,e_{\ell}\rangle)$ for
some $\ell\leq n$, then one can regard $\mathcal{C}$ as a rank-metric subcode
of $\mathbb{F}_{q}^{\ell\times m}$ by deleting the last $n-\ell$ rows of each
matrix. Moreover, the $q$-polymatroid $(\mathbb{F}_{q}^{m},\rho_{r})$ is left
unchanged by this operation and the $q$-polymatroid
$(\mathbb{F}_{q}^{\ell},\rho_{c})$ determines the $q$-polymatroid
$(\mathbb{F}_{q}^{n},\rho_{c})$ according to the formula above.

The next theorem is the main result of this section. We compute the rank
functions of the $q$-polymatroids associated to a quasi-optimal anticode.

######  Theorem 5.5.

Let $h\geq 0$, $0\leq k\leq m$, and $0\leq s\leq m-h$. Let $V=\langle
e_{1},\ldots,e_{s}\rangle,V^{\prime}=\langle
e_{1},\ldots,e_{s+h}\rangle\subseteq\mathbb{F}_{q}^{n}$ and let $U=\langle
e_{1},\ldots,e_{k}\rangle\subseteq\mathbb{F}_{q}^{m}$. Let
$\mathcal{C}=\mbox{Mat}(V)+(\mbox{Mat}(U)^{t}\cap\mbox{Mat}(V^{\prime}))\subseteq\mathbb{F}_{q}^{\,n\times
m}$ be a rank-metric code. Let $J\subseteq\mathbb{F}_{q}^{n}$ and
$K\subseteq\mathbb{F}_{q}^{m}$. The rank functions of the $q$-polymatroids
associated to $\mathcal{C}$ are

$\rho_{r}(\mathcal{C},K)=\frac{h(k-\dim(U\cap K^{\perp}))+s\cdot\dim(K)}{n}$

and

$\rho_{c}(\mathcal{C},J)=s-\dim(V\cap J^{\perp})+\frac{k(h+\dim(V\cap
J^{\perp})-\dim(V^{\prime}\cap J^{\perp}))}{m}.$

###### Proof.

For any $J\subseteq\mathbb{F}_{q}^{n}$, one has

$\mathcal{C}\cap\mbox{Mat}(J)=\mathcal{C}\cap\mbox{Mat}(J\cap V^{\prime})$
(12)

by Lemma 5.4, since $\mathcal{C}\subseteq\mbox{Mat}(V^{\prime})$. Write $J\cap
V^{\prime}$ as $(J\cap V)+J^{\prime}$, where $J^{\prime}\cap V=0$. This can
always be done by letting $J^{\prime}$ be the vector space generated by a set
of vectors that, together with a basis of $J\cap V$, form a basis of $J\cap
V^{\prime}$. Then

$\mathcal{C}\cap\mbox{Mat}(J\cap V^{\prime})=\mathcal{C}\cap\mbox{Mat}(J\cap
V+J^{\prime})=\mathcal{C}\cap[\mbox{Mat}(J\cap
V)+\mbox{Mat}(J^{\prime})]=\mbox{Mat}(J\cap
V)+\mathcal{C}\cap\mbox{Mat}(J^{\prime}).$ (13)

Since $J^{\prime}\cap V=0$, then

$\dim(\mbox{Mat}(J\cap
V)+\mathcal{C}\cap\mbox{Mat}(J^{\prime}))=\dim(\mbox{Mat}(J\cap
V))+\dim(\mathcal{C}\cap\mbox{Mat}(J^{\prime})).$ (14)

Moreover $\dim(\mathcal{C}\cap\mbox{Mat}(J^{\prime}))=k\cdot\dim(J^{\prime})$
by Lemma 5.3. Combining (12), (13), and (14) one gets

$\dim(\mathcal{C}\cap\mbox{Mat}(J))=m\cdot\dim(J\cap
V)+k\cdot\dim(J^{\prime})=m\cdot\dim(J\cap V)+k(\dim(J\cap
V^{\prime})-\dim(J\cap V)).$

Therefore

$\rho_{c}(\mathcal{C},J)=\frac{\dim(\mathcal{C})-\dim(\mathcal{C}\cap\mbox{Mat}(J^{\perp}))}{m}=s-\dim(V\cap
J^{\perp})+\frac{k(h+\dim(V\cap J^{\perp})-\dim(V^{\prime}\cap
J^{\perp}))}{m}$

as claimed. The other equality is proved similarly. ∎

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-PH-EP-2013-176 LHCb-PAPER-2013-053 18 December 2013

Measurement of $D^{0}$–$\kern 4.14793pt\overline{\kern-4.14793ptD}{}^{0}$
mixing parameters and search for $C\\!P$ violation using $D^{0}\rightarrow
K^{+}\pi^{-}$ decays

The LHCb collaboration†††Authors are listed on the following pages.

Measurements of charm mixing parameters from the decay-time-dependent ratio of
$D^{0}\rightarrow K^{+}\pi^{-}$ to $D^{0}\rightarrow K^{-}\pi^{+}$ rates and
the charge-conjugate ratio are reported. The analysis uses data, corresponding
to $3\mbox{\,fb}^{-1}$ of integrated luminosity, from proton-proton collisions
at 7 and 8 $\mathrm{\,Te\kern-1.00006ptV}$ center-of-mass energies recorded by
the LHCb experiment. In the limit of charge-parity ($C\\!P$) symmetry, the
mixing parameters are determined to be $x^{\prime 2}=(5.5\pm 4.9)\times
10^{-5}$, $y^{\prime}=(4.8\pm 1.0)\times 10^{-3}$, and $R_{D}=(3.568\pm
0.066)\times 10^{-3}$. Allowing for $C\\!P$ violation, the mixing parameters
are determined separately for $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons yielding $A_{D}=(-0.7\pm
1.9)\%$, for the direct $C\\!P$-violating asymmetry, and $0.75<|q/p|<1.24$ at
the $68.3\%$ confidence level, where $q$ and $p$ are parameters that describe
the mass eigenstates of the neutral charm mesons in terms of the flavor
eigenstates. This is the most precise determination of these parameters from a
single experiment and shows no evidence for $C\\!P$ violation.

© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.

LHCb collaboration

R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y.
Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58,
E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J. Back47, A.
Badalov35, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw10, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, A. Bursche39, G. Busetto21,q, J. Buytaert37, S. Cadeddu15, O.
Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D.
Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A.
Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51,
L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles54, Ph.
Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37,
X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46,
J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A.
Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G.
Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz Torres59, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F.
Dordei11, P. Dornan52, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, P. Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S.
Easo48, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S.
Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch.
Elsasser39, A. Falabella14,e, C. Färber11, C. Farinelli40, S. Farry51, D.
Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S.
Filippov32, M. Fiore16,e, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i,
R. Forty37, O. Francisco2, M. Frank37, C. Frei37, M. Frosini17,37,f, E.
Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58,
Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C.
Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47, Ph. Ghez4,
V. Gibson46, L. Giubega28, V.V. Gligorov37, C. Göbel59, D. Golubkov30, A.
Golutvin52,30,37, A. Gomes2, P. Gorbounov30,37, H. Gordon37, M. Grabalosa
Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G.
Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, L.
Grillo11, O. Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, B.
Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45,
J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, M. Heß60, A.
Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, W. Hulsbergen40,
P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50, V.
Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E. Jans40,
P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R. Jones46, C.
Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M. Karacson37, T.M.
Karbach37, I.R. Kenyon44, T. Ketel41, B. Khanji20, O. Kochebina7, I.
Komarov38, R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L.
Kravchuk32, K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9,
M. Kucharczyk20,25,37,j, V. Kudryavtsev33, K. Kurek27, T. Kvaratskheliya30,37,
V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W.
Lambert41, E. Lanciotti37, G. Lanfranchi18, C. Langenbruch37, T. Latham47, C.
Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A.
Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11,
Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37,
S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D.
Lucchesi21,q, J. Luisier38, H. Luo49, O. Lupton54, F. Machefert7, I.V.
Machikhiliyan30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G.
Mancinelli6, J. Maratas5, U. Marconi14, P. Marino22,s, R. Märki38, J. Marks11,
G. Martellotti24, A. Martens8, A. Martín Sánchez7, M. Martinelli40, D.
Martinez Santos41,37, D. Martins Tostes2, A. Martynov31, A. Massafferri1, R.
Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A. Mazurov16,37,e, J.
McCarthy44, A. McNab53, R. McNulty12, B. McSkelly51, B. Meadows56,54, F.
Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, A. Mordà6, M.J.
Morello22,s, R. Mountain58, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28,
B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1,
M. Needham49, S. Neubert37, N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C.
Nguyen-Mau38,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A.
Nomerotski54, A. Novoselov34, A. Oblakowska-Mucha26, V. Obraztsov34, S.
Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M.
Otalora Goicochea2, P. Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, M.
Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C.
Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pearce53, A.
Pellegrino40, G. Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez
Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L.
Pescatore44, E. Pesen61, G. Pessina20, K. Petridis52, A. Petrolini19,i, A.
Phan58, E. Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S.
Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E.
Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C. Potterat35, A. Powell54,
J. Prisciandaro38, A. Pritchard51, C. Prouve7, V. Pugatch43, A. Puig
Navarro38, G. Punzi22,r, W. Qian4, B. Rachwal25, J.H. Rademacker45, B.
Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41,
S. Redford54, S. Reichert53, M.M. Reid47, A.C. dos Reis1, S. Ricciardi48, A.
Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5, P. Robbe7, D.A.
Roberts57, A.B. Rodrigues1, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, M. Rotondo21, J. Rouvinet38, T. Ruf37, F.
Ruffini22, H. Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36,
N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, B. Sanmartin
Sedes36, R. Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M.
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Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M. Shapkin34, I.
Shapoval16,42,e, Y. Shcheglov29, T. Shears51, L. Shekhtman33, O. Shevchenko42,
V. Shevchenko30, A. Shires9, R. Silva Coutinho47, M. Sirendi46, N. Skidmore45,
T. Skwarnicki58, N.A. Smith51, E. Smith54,48, E. Smith52, J. Smith46, M.
Smith53, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro38, D. Souza45, B. Souza De
Paula2, B. Spaan9, A. Sparkes49, P. Spradlin50, F. Stagni37, S. Stahl11, O.
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Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M. Veltri17,g,
G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A. Vorobyev29, V.
Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47, R. Wallace12, S.
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Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wimberley57, J. Wishahi9, W. Wislicki27, M. Witek25, G. Wormser7, S.A.
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X. Yuan3, O. Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L.
Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A.
Zvyagin37.

1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
61Celal Bayar University, Manisa, Turkey, associated to 37
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy

Mass eigenstates of neutral charm mesons are linear combinations of flavor
eigenstates $|D_{1,2}\rangle=p|D^{0}\rangle\pm q|\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rangle$, where $p$ and $q$ are
complex parameters. This results in $D^{0}$–$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillation. In the limit of
charge-parity ($C\\!P$) symmetry, the oscillation is characterized by the
difference in mass $\Delta m\equiv m_{2}-m_{1}$ and decay width
$\Delta\Gamma\equiv\Gamma_{2}-\Gamma_{1}$ between the $D$ mass eigenstates.
These differences are usually expressed in terms of the dimensionless mixing
parameters $x\equiv\Delta m/\Gamma$ and $y\equiv\Delta\Gamma/2\Gamma$, where
$\Gamma$ is the average decay width of neutral $D$ mesons. If $C\\!P$ symmetry
is violated, the oscillation rates for mesons produced as $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ can differ, further enriching the
phenomenology. Both short- and long-distance components of the amplitude
contribute to the time evolution of neutral $D$ mesons [1, 2, 3]. Short-
distance amplitudes could include contributions from non-standard-model
particles or interactions, possibly enhancing the average oscillation rate or
the difference between $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ meson rates. The study of $C\\!P$
violation in $D^{0}$ oscillation may lead to an improved understanding of
possible dynamics beyond the standard model [4, 5, 6, 7].

The first evidence for $D^{0}$–$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ oscillation was reported in 2007
[8, 9]. By 2009, the hypothesis of no oscillation was excluded with
significance in excess of 10 standard deviations [10] by combining results
from different experiments [8, 9, 11, 12, 13, 14, 15, 16, 17]. In 2012, the
LHCb experiment reported the first observation from a single measurement with
greater than 5 standard deviation significance [18], which has been recently
confirmed by the CDF experiment [19].

This Letter reports a search for $C\\!P$ violation in $D^{0}$–$\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mixing by comparing the decay-
time-dependent ratio of $D^{0}\rightarrow K^{+}\pi^{-}$ to $D^{0}\rightarrow
K^{-}\pi^{+}$ rates with the corresponding ratio for the charge-conjugate
processes. An improved determination of the $C\\!P$-averaged charm mixing
parameters with respect to our previous measurement [18] is also reported. The
analysis uses data corresponding to $1.0\mbox{\,fb}^{-1}$ of integrated
luminosity from $\sqrt{s}=7\,\mathrm{\,Te\kern-1.00006ptV}$ $pp$ collisions
recorded by LHCb during 2011 and $2.0\mbox{\,fb}^{-1}$ from
$\sqrt{s}=8\,\mathrm{\,Te\kern-1.00006ptV}$ collisions recorded during 2012.
The neutral $D$ flavor at production is determined from the charge of the low-
momentum pion $\pi_{\rm s}^{+}$ in the flavor-conserving strong-interaction
decay $D^{*+}\rightarrow D^{0}\pi_{\rm s}^{+}$. The inclusion of charge-
conjugate processes is implicit unless stated otherwise. The
$D^{*+}\rightarrow D^{0}(\rightarrow K^{-}\pi^{+})\pi_{\rm s}^{+}$ process is
denoted as right sign (RS), and $D^{*+}\rightarrow D^{0}(\rightarrow
K^{+}\pi^{-})\pi_{\rm s}^{+}$ is denoted as wrong sign (WS). The RS decay rate
is dominated by a Cabibbo-favored amplitude. The WS rate arises from the
interfering amplitudes of the doubly Cabibbo-suppressed $D^{0}\rightarrow
K^{+}\pi^{-}$ decay and the Cabibbo-favored $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ decay
following $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
oscillation, each of similar magnitude. In the limit of $|x|,|y|\ll 1$, and
assuming negligible $C\\!P$ violation, the time-dependent ratio $R(t)$ of WS-
to-RS decay rates is [1, 2, 3, 4]

$R(t)\approx R_{D}+\sqrt{R_{D}}\ y^{\prime}\ \frac{t}{\tau}+\frac{x^{\prime
2}+y^{\prime 2}}{4}\left(\frac{t}{\tau}\right)^{2},$ (1)

where $t$ is the decay time, $\tau$ is the average $D^{0}$ lifetime, and
$R_{D}$ is the ratio of suppressed-to-favored decay rates. The parameters
$x^{\prime}$ and $y^{\prime}$ depend linearly on the mixing parameters as
$x^{\prime}\equiv x\cos\delta+y\sin\delta$ and $y^{\prime}\equiv
y\cos\delta-x\sin\delta$, where $\delta$ is the strong-phase difference
between the suppressed and favored amplitudes $\mathcal{A}(D^{0}\rightarrow
K^{+}\pi^{-})/\mathcal{A}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
K^{+}\pi^{-})=-\sqrt{R_{D}}e^{-i\delta}$. Allowing for $C\\!P$ violation, the
WS rates $R^{+}(t)$ and $R^{-}(t)$ of initially produced $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons are functions of
independent sets of mixing parameters $(R_{D}^{\pm},\,x^{\prime
2\pm},\,y^{\prime\pm})$. A difference between $R^{+}_{D}$ and $R^{-}_{D}$
arises if the ratio between the magnitudes of suppressed and favored decay
amplitudes is not $C\\!P$ symmetric (direct $C\\!P$ violation). Violation of
$C\\!P$ symmetry either in mixing $|q/p|\neq 1$ or in the interference between
mixing and decay amplitudes $\phi\equiv\arg\left[q\mathcal{A}(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow
K^{+}\pi^{-})/p\mathcal{A}(D^{0}\rightarrow K^{+}\pi^{-})\right]-\delta\neq 0$
are usually referred to as indirect $C\\!P$ violation and would result in
differences between $(x^{\prime 2+},\,y^{\prime+})$ and $(x^{\prime
2-},\,y^{\prime-})$.

The LHCb detector [20] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, designed for the study of particles
containing $b$ or $c$ quarks. Detector components particularly relevant for
this analysis are the silicon vertex detector, which provides reconstruction
of displaced vertices of $b$\- and $c$-hadron decays; the tracking system,
which measures charged particle momenta with relative uncertainty that varies
from $0.4\%$ at $5\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to $0.6\%$ at
$100\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, corresponding to a typical mass
resolution of approximately $8\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
a two-body charm-meson decay; and the ring-imaging Cherenkov detectors, which
provide kaon-pion discrimination [21]. The magnet polarity is periodically
inverted and approximately equal amounts of data are collected in each
configuration to mitigate the effects of detection asymmetries. The online
event-selection system (trigger) [22] consists of a first-level hardware stage
based on information from the calorimeter and muon systems, followed by a
software high-level trigger.

Events with $D^{*+}$ candidates consistent with being produced at the $pp$
collision point (primary vertex) are selected following Ref. [18]. In
addition, a WS candidate is discarded if resulting from a $D^{0}$ candidate
that, associated with another pion, also forms a RS candidate with
$M(D^{0}\pi_{\rm s}^{+})$ within $3\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
of the known $D^{*+}$ mass. This removes about 15% of the WS background with
negligible signal loss. The two-body $D^{0}\pi_{\rm s}^{+}$ mass
$M(D^{0}\pi_{\rm s}^{+})$ is computed using the known $D^{0}$ and $\pi^{+}$
masses [23] and their reconstructed momenta [18]. In Ref. [18], we used events
selected by the hardware trigger based on hadron calorimeter transverse-energy
depositions that were geometrically matched with signal final-state tracks. In
the present analysis, we distinguish two trigger categories. One category
consists of events that meet the above trigger requirement (triggered-on-
signal, TOS). The other comprises events with candidates failing the track-
calorimeter matching and events selected based on muon hardware triggers
decisions ($\overline{\text{TOS}}$). The two subsamples contribute
approximately equal signal yields with similar purities. However, they require
separate treatment due to their differing kinematic distributions and trigger-
induced biases.

Figure 1: Distribution of $M(D^{0}\pi_{\rm s}^{+})$ for selected (a) right-
sign $D^{0}\rightarrow K^{-}\pi^{+}$ and (b) wrong-sign $D^{0}\rightarrow
K^{+}\pi^{-}$ candidates.

The RS and WS signal yields are determined by fitting the $M(D^{0}\pi_{\rm
s}^{+})$ distribution of $D^{0}$ candidates with reconstructed mass within
$24\,{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the known value. The time-
integrated $M(D^{0}\pi_{\rm s}^{+})$ distributions are shown in Fig. 1. The
smooth background is dominated by favored $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ decays
associated with random $\pi_{\rm s}^{+}$ candidates. The sample contains
$1.15\times 10^{5}$ ($1.14\times 10^{5}$) signal WS $D^{0}$ ($\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$) decays and approximately 230
times more RS decays. Yield differences between $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays are dominated by
differences in charm-anticharm production rates and reconstruction
efficiencies. Each sample is divided into 13 subsamples according to the
candidate’s decay time, and signal yields are determined for each using shape
parametrizations determined from simulation and tuned to data [18]. We assume
that for a given $D^{*}$ meson flavor, the signal shapes are common to WS and
RS decays, while the descriptions of the background can differ. The decay-
time-dependent WS-to-RS yield ratios $R^{+}$ and $R^{-}$ observed in the
$D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ samples,
respectively, and their difference are shown in Fig. 2. These are corrected
for the relative efficiencies for reconstructing $K^{-}\pi^{+}$ and
$K^{+}\pi^{-}$ final states.

Figure 2: Efficiency-corrected ratios of WS-to-RS yields for (a) $D^{*+}$
decays, (b) $D^{*-}$ decays, and (c) their differences as functions of decay
time in units of $D^{0}$ lifetime. Projections of fits allowing for (dashed
line) no $C\\!P$ violation, (dotted line) no direct $C\\!P$ violation, and
(solid line) full $C\\!P$ violation are overlaid. The abscissa of the data
points corresponds to the average decay time over the bin; the error bars
indicate the statistical uncertainties.

The mixing parameters are determined by minimizing a $\chi^{2}$ variable that
includes terms for the difference between the observed and predicted ratios
and for systematic deviations of parameters

$\chi^{2}=\sum_{i}\left[\left(\frac{r^{+}_{i}-\epsilon_{r}^{+}\widetilde{R}_{i}^{+}}{\sigma_{i}^{+}}\right)^{2}+\left(\frac{r^{-}_{i}-\epsilon_{r}^{-}\widetilde{R}_{i}^{-}}{\sigma_{i}^{-}}\right)^{2}\right]+\chi^{2}_{\epsilon}\,+\,\chi^{2}_{B}+\,\chi^{2}_{\rm
p}\,.$ (2)

The measured WS-to-RS yield ratio and its statistical uncertainty in the
decay-time bin $i$ are denoted by $r_{i}^{\pm}$ and $\sigma_{i}^{\pm}$,
respectively. The predicted value for the WS-to-RS yield ratio
$\widetilde{R}_{i}^{\pm}$ corresponds to the time integral over bin $i$ of Eq.
(1) including bin-specific corrections. These account for small biases due to
the decay-time evolution of the approximately $3\%$ fraction of signal
candidates originating from $b$-hadron decays ($\Delta_{B}$) and of the about
$0.5\%$ component of peaking background from RS decays in which both final-
state particles are misidentified ($\Delta_{\rm p}$) [18]. The relative
efficiency $\epsilon_{r}^{\pm}$ accounts for instrumental asymmetries in the
$K\pi$ reconstruction efficiencies, mainly caused by $K^{-}$ mesons having a
larger interaction cross section with matter than $K^{+}$ mesons. These
asymmetries are measured in data to be in the range $0.8$–$1.2\%$ with $0.2\%$
precision and to be independent of decay time. They are derived from the
efficiency ratio
$\epsilon_{r}^{+}=1/\epsilon_{r}^{-}=\epsilon(K^{+}\pi^{-})/\epsilon(K^{-}\pi^{+})$,
obtained from the product of $D^{-}\rightarrow K^{+}\pi^{-}\pi^{-}$ and
$D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}(\rightarrow\pi^{+}\pi^{-})\pi^{+}$ event yields divided by the product of
the corresponding charge-conjugate decay yields. No $C\\!P$ violation is
expected or experimentally observed [23] in these decays. Asymmetries due to
$C\\!P$ violation in neutral kaons and their interaction cross-sections with
matter are negligible. The $1\%$ asymmetry between $D^{+}$ and $D^{-}$
production rates [24] cancels in this ratio, provided that the kinematic
distributions are consistent across samples. We weight the $D^{-}\rightarrow
K^{+}\pi^{-}\pi^{-}$ events so that their kinematic distributions match those
in the $D^{+}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{+}$ sample.
Similarly, these samples are weighted as functions of $K\pi$ momentum to match
the RS momentum spectra. The parameters associated with $\Delta_{B}$,
$\Delta_{\rm p}$, and $\epsilon_{r}$ are determined separately for TOS and
$\overline{\text{TOS}}$ subsets and vary independently in the fit within their
Gaussian constraints $\chi^{2}_{B}$, $\chi^{2}_{\rm p}$, and
$\chi^{2}_{\epsilon}$ [18].

To avoid experimenters’ bias in the $C\\!P$ violation parameters, the
measurement technique is finalized by adding arbitrary offsets to the WS-to-RS
yield ratios for the $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ samples, designed to mimic the
effect of different mixing parameters in the two samples. To rule out global
systematic uncertainties not accounted for in Eq. (2), the data are first
integrated over the whole decay-time spectrum and subsequently divided into
statistically independent subsets according to criteria likely to reveal
biases from specific instrumental effects. These include the number of primary
vertices in the events, the $K$ laboratory momentum, the $\pi_{\rm s}$ impact
parameter $\chi^{2}$ with respect to the primary vertex, the $D^{0}$ impact
parameter $\chi^{2}$ with respect to the primary vertex, the magnetic field
orientation, and the hardware trigger category. The variations of the time-
integrated charge asymmetry in WS-to-RS yield ratios are consistent with
statistical fluctuations. Then, we investigate decay-time-dependent biases by
dividing the time-binned sample according to the magnet polarity and the
number of primary vertices per event. In the TOS sample, differences of WS-to-
RS yield ratios as functions of decay time for opposite magnet polarities
yield $\chi^{2}$ values of 12, 17, and 14 (for 12 degrees of freedom), for
events with one, two, and more than two primary vertices, respectively. The
corresponding $\chi^{2}$ values in the $\overline{\text{TOS}}$ sample, 9, 11,
and 8, suggest a systematically better consistency. Hence, the statistical
uncertainty of each of the WS-to-RS ratios in the TOS samples is increased by
a factor of $\sqrt{17/12}$, following Ref. [23]. These scaled uncertainties
are used in all subsequent fits. Independent analyses of the 2011 and 2012
data yield consistent results. The ratio between RS $D^{0}$ to $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decay rates is independent of
decay time with a $62\%$ $p$ value and a standard deviation of $0.16\%$,
showing no evidence of correlations between particle identification or
reconstruction efficiency and decay time.

Table 1: Results of fits to the data for different hypotheses on the $C\\!P$ symmetry. The reported uncertainties are statistical and systematic, respectively; ndf indicates the number of degrees of freedom. See App. A for fits results including correlation coefficients. Parameter | Value
---|---
Direct and indirect $C\\!P$ violation
$R_{D}^{+}$ [ |  $10^{-3}$] |  $3.545$ $\pm$ |  $0.082$ $\pm$ | $0.048$
$y^{\prime+}$ [ |  $10^{-3}$] |  $5.1$ $\pm$ |  $1.2$ $\pm$ | $0.7$
$x^{\prime 2+}$ [ |  $10^{-5}$] |  $4.9$ $\pm$ |  $6.0$ $\pm$ | $3.6$
$R_{D}^{-}$ [ |  $10^{-3}$] |  $3.591$ $\pm$ |  $0.081$ $\pm$ | $0.048$
$y^{\prime-}$ [ |  $10^{-3}$] |  $4.5$ $\pm$ |  $1.2$ $\pm$ | $0.7$
$x^{\prime 2-}$ [ |  $10^{-5}$] |  $6.0$ $\pm$ |  $5.8$ $\pm$ | $3.6$
$\chi^{2}/\text{ndf}$ | $85.9/98$
No direct $C\\!P$ violation
$R_{D}$ [ |  $10^{-3}$] |  $3.568$ $\pm$ |  $0.058$ $\pm$ | $0.033$
$y^{\prime+}$ [ |  $10^{-3}$] |  $4.8$ $\pm$ |  $0.9$ $\pm$ | $0.6$
$x^{\prime 2+}$ [ |  $10^{-5}$] |  $6.4$ $\pm$ |  $4.7$ $\pm$ | $3.0$
$y^{\prime-}$ [ |  $10^{-3}$] |  $4.8$ $\pm$ |  $0.9$ $\pm$ | $0.6$
$x^{\prime 2-}$ [ |  $10^{-5}$] |  $4.6$ $\pm$ |  $4.6$ $\pm$ | $3.0$
$\chi^{2}/\text{ndf}$ | $86.0/99$
No $C\\!P$ violation
$R_{D}$ [ |  $10^{-3}$] |  $3.568$ $\pm$ |  $0.058$ $\pm$ | $0.033$
$y^{\prime}$ [ |  $10^{-3}$] |  $4.8$ $\pm$ |  $0.8$ $\pm$ | $0.5$
$x^{\prime 2}$ [ |  $10^{-5}$] |  $5.5$ $\pm$ |  $4.2$ $\pm$ | $2.6$
$\chi^{2}/\text{ndf}$ | $86.4/101$

Three fits are performed to the data shown in Fig. 2. The first allows direct
and indirect $C\\!P$ violation; the second allows only indirect $C\\!P$
violation by constraining $R_{D}^{\pm}$ to a common value; and the third is a
$C\\!P$-conserving fit that constrains all mixing parameters to be the same in
the $D^{0}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ samples.
The fit results and their projections are shown in Table 1 and Fig. 2,
respectively. Figure 3 shows the central values and confidence regions in the
$(x^{\prime 2},\,y^{\prime})$ plane. For each fit, 104 WS-to-RS ratio data
points are used, corresponding to 13 ranges of decay time, distinguishing
$D^{*+}$ from $D^{*-}$ decays, TOS from $\overline{\text{TOS}}$ decays, and
2011 data from 2012 data. The consistency with the hypothesis of $C\\!P$
symmetry is determined from the change in $\chi^{2}$ between the fit without
and with $C\\!P$ violation, taking into account the difference in number of
degrees of freedom. The resulting $p$ value, for the fit with direct and
indirect (indirect only) $C\\!P$ violation allowed, is $91\%$ ($81\%$),
showing that the data are compatible with $C\\!P$ symmetry.

Figure 3: Two-dimensional confidence regions in the $(x^{\prime
2},y^{\prime})$ plane obtained (a) without any restriction on $C\\!P$
violation, (b) assuming no direct $C\\!P$ violation, and (c) assuming $C\\!P$
conservation. The dashed (solid) curves in (a) and (b) indicate the contours
of the mixing parameters associated with $\kern
1.79997pt\overline{\kern-1.79997ptD}{}^{0}$ ($D^{0}$) decays. The best-fit
value for $\kern 1.79997pt\overline{\kern-1.79997ptD}{}^{0}$ ($D^{0}$) decays
is shown with an open (filled) point. The solid, dashed, and dotted curves in
(c) indicate the contours of $C\\!P$-averaged mixing parameters at 68.3%,
95.5%, and 99.7% confidence level (CL), respectively. The best-fit value is
shown with a point.

The uncertainties incorporate both statistical and systematic contributions,
since all relevant systematic effects depend on the true values of the mixing
parameters, and are thus incorporated into the fit $\chi^{2}$. These include
the uncertainty in the fraction of charm mesons from $b$-hadron decays, and
their bias on the observed decay time; the uncertainty in the fraction of
peaking background; and the uncertainty in the determination of the
instrumental asymmetry. The statistical uncertainty is determined in a
separate fit and used to calculate the systematic component by subtraction in
quadrature.

Direct $C\\!P$ violation would produce a nonzero intercept at $t=0$ in the
efficiency-corrected difference of WS-to-RS yield ratios between $D^{0}$ and
$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ mesons shown in Fig. 2 (c).
It is parametrized by the asymmetry measured in the first fit
$A_{D}\equiv{(R_{D}^{+}-R_{D}^{-})/(R_{D}^{+}+R_{D}^{-})}=(-0.7\pm 1.9)\%$.
Indirect $C\\!P$ violation results in a time dependence of the efficiency-
corrected difference of yield ratios. The slope observed in Fig. 2 (c) is
about 5% of the individual slopes of Figs. 2 (a) and (b) and is consistent
with zero. From the results of the fit allowing for direct and indirect
$C\\!P$ violation, a likelihood for $|q/p|$ is constructed using the relations
$x^{\prime\pm}=|q/p|^{\pm 1}(x^{\prime}\cos\phi\pm y^{\prime}\sin\phi)$ and
$y^{\prime\pm}=|q/p|^{\pm 1}(y^{\prime}\cos\phi\mp x^{\prime}\sin\phi)$.
Confidence intervals are derived with a likelihood-ratio ordering and assuming
that the correlations are independent of the true values of the mixing
parameters. The magnitude of $q/p$ is determined to be $0.75<|q/p|<1.24$ and
$0.67<|q/p|<1.52$ at the $68.3\%$ and $95.5\%$ confidence levels,
respectively. Significantly more stringent bounds on $|q/p|$ and additional
information on $\phi$ are available by combining the present results with
other measurements [10], in particular when also using theoretical
constraints, such as the relationship $\tan\phi=x(1-|q/p|^{2})/y(1+|q/p|^{2})$
[25, 26], which applies in the limit that direct $C\\!P$ violation is
negligible.

In summary, $D^{0}$–$\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}$
oscillation is studied using $D^{*+}\rightarrow D^{0}(\rightarrow
K^{+}\pi^{-})\pi^{+}$ decays reconstructed in the full sample of $pp$
collisions, corresponding to $3\mbox{\,fb}^{-1}$ of integrated luminosity
collected by the LHCb experiment in 2011 and 2012. Assuming $C\\!P$
conservation, the mixing parameters are measured to be $x^{\prime 2}=(5.5\pm
4.9)\times 10^{-5}$, $y^{\prime}=(4.8\pm 1.0)\times 10^{-3}$, and
$R_{D}=(3.568\pm 0.066)\times 10^{-3}$. The observed parameters are consistent
with, $2.5$ times more precise than, and supersede the results based on a
subset of the present data [18]. Studying $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays separately shows no
evidence for $C\\!P$ violation and provides the most stringent bounds on the
parameters $A_{D}$ and $|q/p|$ from a single experiment.

## Acknowledgements

We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.

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##  Appendix A Supplemental material

Table 2: Detailed fit results. Reported uncertainties and correlation
coefficients include both statistical and systematic sources.

Direct and indirect $C\\!P$ violation
---
Results | Correlations
Parameter | Fit value | $R_{D}^{+}$ | $y^{\prime+}$ | $x^{\prime 2+}$ | $R_{D}^{-}$ | $y^{\prime-}$ | $x^{\prime 2-}$
$R_{D}^{+}$ [ |  $10^{-3}$] | $3.545\pm 0.095$ | $1.000$ | $-0.942$ | $\phantom{-}0.862$ | $-0.016$ | $-0.007$ | $\phantom{-}0.006$
$y^{\prime+}$ [ |  $10^{-3}$] | $5.1\pm 1.4$ |  | $\phantom{-}1.000$ | $-0.968$ | $-0.007$ | $\phantom{-}0.007$ | $-0.007$
$x^{\prime 2+}$ [ |  $10^{-5}$] | $4.9\pm 7.0$ |  |  | $\phantom{-}1.000$ | $\phantom{-}0.005$ | $-0.007$ | $\phantom{-}0.008$
$R_{D}^{-}$ [ |  $10^{-3}$] | $3.591\pm 0.094$ |  |  |  | $\phantom{-}1.000$ | $-0.941$ | $\phantom{-}0.858$
$y^{\prime-}$ [ |  $10^{-3}$] | $4.5\pm 1.4$ |  |  |  |  | $\phantom{-}1.000$ | $-0.966$
$x^{\prime 2-}$ [ |  $10^{-5}$] | $6.0\pm 6.8$ |  |  |  |  |  | $\phantom{-}1.000$

No direct $C\\!P$ violation
---
Results | Correlations
Parameter | Fit value | $R_{D}$ | $y^{\prime+}$ | $x^{\prime 2+}$ | $y^{\prime-}$ | $x^{\prime 2-}$
$R_{D}$ [ |  $10^{-3}$] | $3.568\pm 0.066$ | $1.000$ | $-0.894$ | $\phantom{-}0.770$ | $-0.895$ | $\phantom{-}0.772$
$y^{\prime+}$ [ |  $10^{-3}$] | $4.8\pm 1.1$ |  | $\phantom{-}1.000$ | $-0.949$ | $\phantom{-}0.765$ | $-0.662$
$x^{\prime 2+}$ [ |  $10^{-5}$] | $6.4\pm 5.5$ |  |  | $\phantom{-}1.000$ | $-0.662$ | $\phantom{-}0.574$
$y^{\prime-}$ [ |  $10^{-3}$] | $4.8\pm 1.1$ |  |  |  | $\phantom{-}1.000$ | $-0.950$
$x^{\prime 2-}$ [ |  $10^{-5}$] | $4.6\pm 5.5$ |  |  |  |  | $\phantom{-}1.000$

No $C\\!P$ violation
---
Results | Correlations
Parameter | Fit value | $R_{D}$ | $y^{\prime}$ | $x^{\prime 2}$
$R_{D}$ [ |  $10^{-3}$] | $3.568\pm 0.066$ | $1.000$ | $-0.953$ | $\phantom{-}0.869$
$y^{\prime}$ [ |  $10^{-3}$] | $4.8\pm 1.0$ |  | $\phantom{-}1.000$ | $-0.967$
$x^{\prime 2}$ [ |  $10^{-5}$] | $5.5\pm 4.9$ |  |  | $\phantom{-}1.000$