# Optimal Selection for Good polynomials of degree up to five

Austin Dukes  University of South Florida
4202 E Fowler Ave
33620 Tampa, US<EMAIL_ADDRESS>,  Andrea Ferraguti  Scuola Normale
Superiore, Piazza dei Cavalieri 7, 56126 Pisa<EMAIL_ADDRESS>and
Giacomo Micheli  University of South Florida
4202 E Fowler Ave
33620 Tampa, US<EMAIL_ADDRESS>
###### Abstract.

Good polynomials are the fundamental objects in the Tamo-Barg constructions of
Locally Recoverable Codes (LRC). In this paper we classify all good
polynomials up to degree $5$, providing explicit bounds on the maximal number
$\ell$ of sets of size $r+1$ where a polynomial of degree $r+1$ is constant,
up to $r=4$. This directly provides an explicit estimate (up to an error term
of $O(\sqrt{q})$, with explict constant) for the maximal length and dimension
of a Tamo-Barg LRC. Moreover, we explain how to construct good polynomials
achieving these bounds. Finally, we provide computational examples to show how
close our estimates are to the actual values of $\ell$, and we explain how to
obtain the best possible good polynomials in degree $5$.

###### Key words and phrases:

Good polynomials; Monodromy groups; Global function fields.

###### 2020 Mathematics Subject Classification:

11T06

##  1\. Introduction

A code $\mathcal{C}$ with length $n$ and dimension $k$ is called a locally
recoverable code (LRC) with locality $r$, or a $(n,k,r)$-LRC, if, for any
$v=(v_{1},\cdots,v_{n})\in\mathcal{C}$ and any $1\leq i\leq n$, the coordinate
$v_{i}$ is a function of at most $r$ other coordinates
$v_{i_{1}},v_{i_{2}},\ldots,v_{i_{r}}$ of $v$. In other words, the value of
any symbol in a particular codeword can be recovered by accessing at most $r$
other symbols of the codeword.

Given the linear $(n,k,r)$-LRC $\mathcal{C}$, Gopalan et al. [12] and
Papailiopoulos and Dimakis [18] proved that the minimum distance
$d(\mathcal{C})$ of $\mathcal{C}$ satisfies the upper bound
$d(\mathcal{C})\leq n-k-\lceil k/r\rceil+2$. As in the literature, we will say
$\mathcal{C}$ is an optimal LRC if the minimum distance $d(\mathcal{C})$ of
$\mathcal{C}$ achieves this bound, that is, if
$d(\mathcal{C})=n-k-\lceil{k/r}\rceil+2$.

A powerful approach to constructing LRCs was given by Tamo and Barg in [20],
and it can be accomplished by constructing polynomials of degree $r+1$ which
are constant on pairwise disjoint subsets of $\mathbb{F}_{q}$ of size $r+1$.
Such polynomials are called good polynomials.

More formally, for a nonnegative integer $\ell$ the polynomial
$f\in\mathbb{F}_{q}[X]$ is said to be $(r,\ell)$-good if

  * •

the degree of $f$ is $r+1$, and

  * •

there are pairwise disjoint sets
$A_{1},\ldots,A_{\ell}\subseteq\mathbb{F}_{q}$, each of cardinality $r+1$,
such that $f(A_{i})=\\{t_{i}\\}$ for some $t_{i}\in\mathbb{F}_{q}$, i.e., $f$
is constant on $A_{i}$.

Given a good polynomial, one can construct an optimal linear LRC as follows
(we use the notation of [13], which is the most convenient for our purposes).
Fix $r\geq 1$, and let $f(X)\in\mathbb{F}_{q}$ be a good polynomial. Write
$n=(r+1)\ell$ and $k=rt$, where $t\leq\ell$. For $a=(a_{ij}\mid
i=0,\ldots,r-1;j=0,\ldots t-1)\in\left(\mathbb{F}_{q}\right)^{k}$, define
encoding polynomials

$f_{a}(X)=\sum_{i=1}^{r-1}\sum_{j=0}^{t-1}a_{ij}f(X)^{j}X^{i}.$

Let $A=\bigcup_{i=1}^{\ell}A_{i}$ and define

$\mathcal{C}=\left\\{(f_{a}(x),x\in A)\mid
a\in\left(\mathbb{F}_{q}\right)^{k}\right\\}.$

Then $\mathcal{C}$ is an optimal linear $(n,k,r)$-LRC code over
$\mathbb{F}_{q}$.

In the rest of the paper we classify all $(r,\ell)$-good polynomials up to
$r=4$ as follows: for any fixed prime power $q$ (even or odd) and a fixed $r$
up to $4$, we provide an explicit estimate (of the form $cq+O(\sqrt{q})$,
where $c\in[0,1)$, and the implied constant in the error term is explicitly
computable) on the maximal $\ell$ such that a polynomial of degree $r+1$ is
$(r,\ell)$-good. Moreover, we provide examples of polynomials achieving these
values for $\ell$, showing that the estimate is the best possible.

The machinery we use involves Galois theory, the classification of transitive
subgroups of the symmetric group $\mathcal{S}_{n}$ up to $n=5$, and the theory
of function fields, using the results and techniques of [15, 16], then further
developed in [1, 2, 4, 5, 8, 9].

##  2\. Monodromy groups and totally split places

Let $q$ be a prime power and $f\in{\mathbb{F}}_{q}[X]$ with
$f\notin{\mathbb{F}}_{q}[X^{p}]$, where $p$ is the characteristic of
${\mathbb{F}}_{q}$. Throughout the paper, we will call such polynomials
separable, for short. Let $t$ be transcendental over ${\mathbb{F}}_{q}$.

######  Definition 2.1.

The _arithmetic monodromy group_ of $f$, denoted by $A(f)$, is the Galois
group of $f(X)-t$ seen as a polynomial over ${\mathbb{F}}_{q}(t)$. The
_geometric monodromy group_ of $f$, denoted by $G(f)$, is the Galois group of
$f(X)-t$ seen as a polynomial over $\overline{{\mathbb{F}}}_{q}(t)$.

It is easy to see for any non-constant $f$, the polynomial $f-t$ is
geometrically irreducible. Hence both the arithmetic and the geometric
monodromy groups are isomorphic to transitive subgroups of the symmetric group
of degree $\deg f$, after choosing a labeling of the roots of $f-t$ in
$\overline{{\mathbb{F}}_{q}(t)}$. Different labelings yield conjugate
embeddings.

Recall that $G(f)$ is a normal subgroup of $A(f)$, and the quotient is
isomorphic to the Galois group of the extension
$(M\cap\overline{{\mathbb{F}}}_{q})/{\mathbb{F}}_{q}$, where $M$ is the
splitting field of $f-t$.

For any fixed $n$, it is possible to construct a polynomial $f$ of degree $n$
having $A(f)=G(f)=S_{n}$. Hence one can define a function

$G_{n}(\cdot)\colon\\{\mbox{prime powers}\\}\to{\mathbb{N}}$

that assigns to every prime power $q$ the least positive integer such that
there exists a separable $f\in{\mathbb{F}}_{q}[X]$ of degree $n$ with
$|G(f)|=|A(f)|=G_{n}(q)$. Notice that $G_{n}(q)\geq n$ for every $q$, because
a transitive group of degree $n$ must have at least order $n$.

Thanks to the the techniques introduced in [16], given a separable polynomial
$f\in{\mathbb{F}}_{q}[X]$ such that $A(f)=G(f)$, one can obtain an explicit
estimate on cardinality of the set

$\mathcal{T}^{1}_{split}(f)\coloneqq\\{t_{0}\in{\mathbb{F}}_{q}\colon
f(X)-t_{0}\mbox{ splits into $\deg f$ distinct linear factors}\\}.$

This is done via the following result:

######  Proposition 2.2.

[16, Proposition 3.1] Let $f\in{\mathbb{F}}_{q}[X]$ be a separable polynomial
of degree $n$ with $G(f)=A(f)$ and let $g_{f}$ be the genus of the splitting
field of $f(X)-t$. Then

$\frac{q+1-2g_{f}\sqrt{q}}{|G(f)|}-\frac{n}{2}\leq\mathcal{|}T^{1}_{split}(f)|\leq\frac{q+1+2g_{f}\sqrt{q}}{|G(f)|}$

The genus $g_{f}$ can be bounded solely in terms of $\deg f$, using for
example Castelnuovo’s inequality. As noticed in [16, Proposition 3.3], if
$\text{char }{\mathbb{F}}_{q}\nmid|G(f)|$ we have
$g_{f}\leq\frac{(n-2)|G(f)|+2}{2}$.

It is clear from the above proposition that for a fixed $n$, minimizing
$|G(f)|$ maximizes the expected number of totally split places, which in turn
maximises the dimension of the Tamo-Barg code.

In this paper, we compute the function $G_{n}$ for every
$n\in\\{2,\ldots,5\\}$. The simpler cases $n=2,3,4$ are completely treated in
Section 3. When $n=5$ the problem becomes more difficult as, up to
conjugation, there are 5 transitive subgroups of the symmetric group $S_{5}$:

  * •

The cyclic group $C_{5}$, generated by a $5$-cycle;

  * •

The dihedral group $D_{5}$, generated by a $5$-cycle and a product of two
disjoint transpositions;

  * •

The affine general linear group $\operatorname{AGL}_{1}({\mathbb{F}}_{5})$,
isomorphic to $C_{5}\rtimes C_{4}$, generated by a $5$-cycle and a $4$-cycle.

  * •

The alternating group $A_{5}$;

  * •

The symmetric group $S_{5}$.

Nevertheless, we are able to prove the following theorem for good polynomials
of degree $5$.

######  Theorem 2.3.

Let $q$ be a prime power. Then:

$G_{5}(q)=\begin{cases}5&\mbox{if }5\mid q(q-1)\\\ 10&\mbox{if }5\mid q+1\\\
120&\mbox{otherwise}\\\ \end{cases}.$

As we mentioned above, the estimate of Proposition 2.2 can be made explicit,
leading to a formula for the maximal dimension of a Tamo-Barg code of locality
$4$. When $G(f)=A(f)\cong D_{5},C_{5}$ the genus of the splitting field of $f$
is necessarily zero (by Hurwitz formula, for example), and the error term is
therefore $O(1)$. In the next theorem we give an explicit estimate for the
remaining cases, restricting for simplicity to the case $2,3,5\nmid q$.

######  Theorem 2.4.

Let $q$ be a prime power with $2,3,5\nmid q$. Let $f\in{\mathbb{F}}_{q}[X]$ of
degree $5$ with $G(f)=A(f)\cong S_{5}$. Then:

$\frac{q+1-72\sqrt{q}}{120}-\frac{5}{2}\leq|\mathcal{T}^{1}_{split}(f)|\leq\frac{q+1+72\sqrt{q}}{120}.$

###### Proof.

Let $x$ be a root of $f(X)-t$. Let $F\coloneqq\overline{{\mathbb{F}}}_{q}(x)$
and $M$ be the splitting field of $f-t$ over $\overline{{\mathbb{F}}}_{q}(t)$.
By Proposition 2.2, all we have to do is bound the genus $g_{M}$ of $M$. We
will do it via Riemann-Hurwitz applied to the degree $24$ extension $M/F$.
Notice that $F$ has genus $g_{F}$ equal to zero. We have that:

$2g_{M}-2=24(2g_{F}-2)+\sum_{P}\sum_{Q|P}(e_{Q|P}-1),$

because thanks to our assumptions on $q$ the ramification is tame. Here the
external sum is over all places $P$ of $F$, while the internal one is over all
places $Q$ of $M$ dividing $P$, and $e(Q|P)$ is the ramification index. Since
$f$ has degree $5$, its derivative has degree $4$ and therefore there are at
most $5$ places of $F$ that can ramify in $M$ (notice that a place of
$\overline{{\mathbb{F}}}_{q}(t)$ ramifies in $M$ if and only if it ramifies in
$F$). Now since $M/F$ is a Galois extension the ramification index $e(Q|P)$
depends only on $P$, and it is at most $24$. On the other hand, there are at
least two places of $F$ that ramify in $M$, since there are at least two
places of $\overline{{\mathbb{F}}}_{q}(t)$ that ramify in $F$: the infinite
place and a finite one, since the derivative of $f$ has positive degree. All
in all, we have that

$\sum_{P}\sum_{Q|P}(e_{Q|P}-1)=\sum_{P}e_{Q|P}-\sum_{P}\sum_{Q|P}1\leq 5\cdot
24-2=118,$

and substituting in the above equation yields $g_{M}\leq 36$. ∎

##  3\. Degrees up to $4$

In this section we compute $G_{2},G_{3}$ and $G_{4}$. We start with two
general lemmas.

######  Lemma 3.1.

Let $p$ be a prime and $q=p^{m}$ for some $m\geq 1$. Let
$f=X^{q}-X\in{\mathbb{F}}_{q}[X]$. Then $A(f)=G(f)\cong(C_{p})^{m}$.

###### Proof.

Let $x$ be a root of $f$ and let $F\coloneqq{\mathbb{F}}_{q}(x)$. Then $F$ is
a Galois extension of ${\mathbb{F}}_{q}(t)$, because $x+\alpha$ is a root of
$f(X)-t$ for every $\alpha\in{\mathbb{F}}_{q}$, and therefore $|A(f)|=q$.
Since $f(X)-t$ is absolutely irreducible, both $A(f)$ and $G(f)$ act
transitively on the set of roots of $f(X)-t$ and therefore it must be that
$G(f)=A(f)$. If $\sigma\in A(f)$ and $r=x+\alpha$ is a root of $f-t$ for some
$\alpha\in{\mathbb{F}}_{q}$, then $\sigma(r)=r+\beta$ for some
$\beta\in{\mathbb{F}}_{q}$, and therefore $\sigma^{p}(r)=r$. It follows that
$\sigma^{p}$ is the identity, and therefore $A(f)=(C_{p})^{m}$. ∎

######  Lemma 3.2.

Let $\ell$ be a prime and $q$ a prime power with $\ell\nmid q$. Let
$f\in{\mathbb{F}}_{q}(X)$ be a degree $\ell$ polynomial. Then $G(f)=A(f)\cong
C_{\ell}$ if and only if $\ell\mid q-1$ and $f=(X-a)^{\ell}+b$ for some
$a,b\in{\mathbb{F}}_{q}$.

###### Proof.

Sufficiency is obvious.

Conversely, suppose that $G(f)=A(f)\cong C_{\ell}$. Let $x$ be a root of
$f(X)-t$ and $F\coloneqq{\mathbb{F}}_{q}(x)$. Then the ramification in $F$ is
always tame, and hence the Riemann-Hurwitz formula implies that there must be
a finite place of $\overline{{\mathbb{F}}}_{q}(t)$ that ramifies in $F$. Let
this place correspond to $b\in\overline{{\mathbb{F}}}_{q}$. Then $f(X)-b$ must
factor as $(X-a)^{\ell}$ for some $a\in\overline{{\mathbb{F}}}_{q}$. Comparing
the coefficients of the linear terms, it follows immediately that
$a\in{\mathbb{F}}_{q}$, and hence $b\in{\mathbb{F}}_{q}$. But then
$f=(X-a)^{\ell}+b$, and in order to have $G(f)=A(f)$ the field of constants
$F\cap\overline{{\mathbb{F}}}_{q}$ must be ${\mathbb{F}}_{q}$, implying
immediately that $\ell\mid q-1$ because certainly $F$ contains a primitive
$\ell$-th root of unity. ∎

######  Theorem 3.3.

The following holds:

  1. (1)

$G_{2}(q)=2$ for every $q$.

  2. (2)

$G_{3}(q)=\begin{cases}3&\mbox{if }3\mid q(q-1)\\\
6&\mbox{otherwise}\end{cases}$.

###### Proof.

When $n=2$ and $q$ is odd, every quadratic $f\in{\mathbb{F}}_{q}[X]$ has
$G(f)=A(f)\cong C_{2}$. When $q$ is even, by Lemma 3.1 if
$f=X^{2}+X\in{\mathbb{F}}_{q}(X)$ we have $G(f)=A(f)=2$.

When $n=3$ and $3\mid q$, by Lemma 3.1 for $f=X^{3}-X$ we have $G(f)=A(f)\cong
C_{3}$. When $3\mid q-1$ for $f=X^{3}$ we have $G(f)=A(f)\cong C_{3}$. When
$3\nmid q(q-1)$ by Lemma 3.2 we cannot have $G(f)=A(f)=C_{3}$. The only other
transitive group inside $S_{3}$ is $S_{3}$ itself, and hence $G_{3}(q)=6$. ∎

######  Theorem 3.4.

$G_{4}(q)=\begin{cases}24&\mbox{if }q=2\\\ 4&\mbox{if $4\mid q-1$ or $q=2^{m}$
for some $m>1$}\\\ 8&\mbox{otherwise}\\\ \end{cases}.$

###### Proof.

Recall that the transitive groups of degree $4$ are: $C_{4},C_{2}\times
C_{2},D_{4},A_{4}$ and $S_{4}$. Here the non-trivial elements of $C_{2}\times
C_{2}$ are products of two disjoint transpositions, and therefore this copy of
$C_{2}\times C_{2}$ is contained in $A_{4}$.

If $q$ is even and greater than $2$, then there exist distinct elements
$\alpha_{1},\alpha_{2},\alpha_{3}\in{\mathbb{F}}_{q}^{*}$ such that
$\alpha_{1}+\alpha_{2}+\alpha_{3}=0$. Now let
$f=X(X+\alpha_{1})(X+\alpha_{2})(X+\alpha_{3})$; this is of the form
$X^{4}+aX^{2}+bX$ for some $a,b$ with $b\neq 0$. Therefore if $x$ is a root of
$f(X)-t$, then all other roots are of the form $x+\alpha_{i}$ for some
$i\in\\{1,2,3\\}$. It follows that ${\mathbb{F}}_{q}(x)/{\mathbb{F}}_{q}(t)$
is a Galois extension of degree $4$, and therefore $G(f)=A(f)$ and
$G_{2}(q)=4$. In fact it is easy to see that these monodromy groups are
isomorphic to $C_{2}\times C_{2}$: if $\sigma$ is any element of $G(f)$ and
$r$ is any root of $f-t$, then $\sigma(r)=r+\alpha$ for some
$\alpha\in{\mathbb{F}}_{q}$, and hence $\sigma^{2}(r)=r$, showing that non-
trivial elements have order $2$.

If $q=2$, a quick search shows that the only two separable polynomials $f$
with $A(f)\neq S_{4}$ are $X^{4}+X$ and $X^{4}+X^{2}+X$. However, the first
one has $A(f)\cong D_{8}$ and $G(f)\cong C_{2}\times C_{2}$, while the second
one has $A(f)\cong A_{4}$ and $G(f)\cong C_{2}\times C_{2}$. Hence
$G_{4}(2)=24$.

If $4\mid q-1$, then for $f=X^{4}$ we have $G(f)=A(f)\cong C_{4}$, and
therefore $G_{4}(q)=4$.

Finally, suppose $q$ is odd and $4\nmid q-1$. If $A(f)\leq A_{4}$, then the
discriminant of $f-t$ is a square in ${\mathbb{F}}_{q}(t)$. However, such
discriminant is always a polynomial of degree $3$ in $t$, so $A(f)\leq A_{4}$
can never happen (and therefore, in particular, we cannot have $A(f)\cong
C_{2}\times C_{2}$). If it was $A(f)=G(f)\cong C_{4}$, then since the
ramification in the splitting field $F$ of $f-t$ is tame, by Riemann-Hurwitz
there is a finite place of $\overline{{\mathbb{F}}}_{q}(t)$ that ramifies in
$F$. Let $R$ be a place of $F$ lying over it. Then the decomposition group
$D(R|P)$ is either $C_{4}$ or $C_{2}$. In the former case, for some
$t_{0}\in\overline{{\mathbb{F}}}_{q}$ the polynomial $f-t_{0}$ factors as
$(X-a)^{4}$, and this leads to a contradiction as in the proof of Lemma 3.2.
In the latter case, up to translations we have $f-t_{0}=X^{2}(X-a)^{2}$ for
some $a,t_{0}\in\overline{{\mathbb{F}}}_{q}$ with $a\neq 0$, because the
element of order $2$ in $C_{4}$ is a product of two disjoint transpositions.
This implies that $f$ equals the composition $g\circ h$, where $g=X^{2}$ and
$h=X(X-a)$, and it is a well-known fact (see for example [10]) that for the
Galois group of $g\circ h-t$ to be smaller than $D_{4}$ one would need $t$ and
$a^{2}/4-t$ to be linearly dependent in the ${\mathbb{F}}_{2}$-vector space
${\mathbb{F}}_{q}(t)^{*}/{{\mathbb{F}}_{q}(t)^{*}}^{2}$, which clearly does
not hold since $a\neq 0$. Hence $G_{4}(q)\geq 8$. On the other hand, again for
the same well-known reasons one has that if $f=X^{4}+bX^{2}$ for some $b\neq
0$ then $G(f)=A(f)=D_{4}$. Hence $G_{4}(q)=8$. ∎

##  4\. Degree $5$: $\operatorname{AGL}_{1}(\mathbb{F}_{5})$ never occurs

In this section we will prove that if $5\nmid q$ then there exists no degree
$5$ polynomial $f$ with $G(f)\cong\operatorname{AGL}_{1}({\mathbb{F}}_{5})$.
From now on, we let $M$ be the splitting field of $f-t$ over
${\mathbb{F}}_{q}(t)$. If $P$ is a place of $\overline{{\mathbb{F}}}_{q}(t)$
and $R$ is a place of $M$ lying above it, we denote by $D(R|P)$ the
corresponding decomposition group. For every
$t_{0}\in\overline{{\mathbb{F}}}_{q}$, we denote by $P_{t_{0}}$ the
corresponding place of $\overline{{\mathbb{F}}}_{q}(t)$.

We start with a preliminary lemma.

######  Lemma 4.1.

Let $q$ be a prime power with $5\nmid q$. Let $f\in{\mathbb{F}}_{q}[X]$ be a
degree 5 polynomial and assume that
$G(f)\cong\operatorname{AGL}_{1}({\mathbb{F}}_{5})$. Then there are
$a,b,t_{0}\in\overline{{\mathbb{F}}}_{q}$ with $a\neq b$ such that
$f-t_{0}=(X-a)^{4}(X-b)$.

###### Proof.

We start by showing that
$M=\overline{\mathbb{F}}_{q}(x,x^{\prime})=\overline{\mathbb{F}}_{q}(x)\overline{\mathbb{F}}_{q}(x^{\prime})$
for any two roots $x\neq x^{\prime}$ of $f(X)-t$ in $M$. Observe that
$\overline{\mathbb{F}}_{q}(t)(x)=\overline{\mathbb{F}}_{q}(x)$ since
$t=f(x)\in\overline{\mathbb{F}}_{q}(x)$ (and similarly for $x^{\prime}$) and
write $F=\overline{\mathbb{F}}_{q}(x)$ and
$F^{\prime}=\overline{\mathbb{F}}_{q}(x^{\prime})$. Clearly we have
$[F:\overline{\mathbb{F}}_{q}(t)]=[F^{\prime}:\overline{\mathbb{F}}_{q}(t)]=\deg
f=5$. Because $G(f)\cong\operatorname{AGL}_{1}(\mathbb{F}_{5})$ is
$2$-transitive, the stabilizer $G_{x}\subseteq G$ of $x$ acts transitively on
the four other roots of $f-t$. In particular, since $G_{x}$ is the Galois
group of $(f(X)-t)/(X-x)$ over $\overline{\mathbb{F}}_{q}(t)$, and since the
orbit of $x^{\prime}$ under the action of $G_{x}$ is a set of size $4$, we
have $[FF^{\prime}:F]=[G:G_{x}]=4$. By definition $M\supseteq FF^{\prime}$, so
since $|G|=20$ we have
$M=FF^{\prime}=\overline{\mathbb{F}}_{q}(x,x^{\prime})$.

Now, let $P_{\infty}$ be the place at infinity of
$\overline{\mathbb{F}}_{q}(t)$ and let $R_{\infty}$ be a place of $M$ lying
over $P_{\infty}$. Let $Q_{\infty}=R_{\infty}\cap F$ and consider the
ramification index $e(Q_{\infty}|P_{\infty})$ of $Q_{\infty}$ over
$P_{\infty}$. Recalling that $f(x)=t$ in $F$, we have

$\upsilon_{Q_{\infty}}(f(x))=\upsilon_{Q_{\infty}}(t)=\upsilon_{P_{\infty}}(t)\cdot
e(Q_{\infty}|P_{\infty})=-e(Q_{\infty}|P_{\infty}),$

since $P_{\infty}$ is a pole of order $1$ of $t$. On the other hand, we also
have $\upsilon_{Q_{\infty}}(f(x))=\deg f\cdot\upsilon_{Q_{\infty}}(x)=-5$ by
the strict triangle inequality. This yields $e(Q_{\infty}|P_{\infty})=5$, and
an identical argument applied to $Q^{\prime}_{\infty}=R_{\infty}\cap
F^{\prime}$ yields $e(Q^{\prime}_{\infty}|P_{\infty})=5$. Since we have seen
that $M$ is the compositum of the fields $F$ and $F^{\prime}$ (both of which
are tame extensions of $\overline{\mathbb{F}}_{q}(t)$ as we are working in
characteristic $\neq 5$), it now follows from Abhyankar’s Lemma [19, Theorem
3.9.1] that
$e(R_{\infty}|P_{\infty})=\textup{lcm}\\{e(Q_{\infty}|P_{\infty}),e(Q^{\prime}_{\infty}|P_{\infty})\\}=5$.
Thus the decomposition group $D(R_{\infty}|P_{\infty})$ is a group of order
$5$, and hence it is isomorphic to $C_{5}$.

Next, we claim that there must be some $t_{0}\in\overline{\mathbb{F}}_{q}$
such that for any place $R$ of $M$ lying over $P_{t_{0}}$, the decomposition
group $D(R|P_{t_{0}})$ is isomorphic to $C_{4}$. To prove it, start by
noticing that there must be some $t_{0}\in\overline{\mathbb{F}}_{q}$ such that
the decomposition group of any place of $M$ lying above it contains a cycle of
order $4$. In fact, consider the subset of $G(f)$ of elements of even order
that belong to some decomposition group: this contains no transpositions
because the transitive copy of $\operatorname{AGL}_{1}({\mathbb{F}}_{5})$
inside $S_{5}$ contains no transpositions, and on the other hand if all such
elements had order $2$ then they would all be products of two transpositions.
However the decomposition groups generate $G(f)$,111This is because if $G$ is
the subgroup of $G(f)$ generated by all the decomposition subgroups, then
$M^{G}$ is an unramified extension of $\overline{{\mathbb{F}}}_{q}(t)$, and
there are no non-trivial such extensions. and in this latter case it would
follow that $G(f)\leq A_{5}$, which is false once again. So let
$t_{0}\in\overline{\mathbb{F}}_{q}$ be such that for some place $R$ of $M$
lying above $P_{t_{0}}$, the decomposition group contains a cycle of order
$4$. If we had $C_{4}\subset D(R|P_{t_{0}})\neq C_{4}$, it would follow
$D(R|P_{t_{0}})\cong G(f)$ by the maximality of $C_{4}$ in $G(f)$. But then
$R|P_{t_{0}}$ would be totally ramified, and hence $f(X)-t_{0}=(X-a_{0})^{5}$
for some $a_{0}\in\overline{\mathbb{F}}_{q}$. Since the field of constants of
$M/\overline{\mathbb{F}}_{q}(t)$ is trivial, this factorization implies $5\mid
q-1$. But then $G\cong C_{5}$, an immediate contradiction. Thus
$D(R|P_{t_{0}})\cong C_{4}$.

To conclude the proof, notice that specializing at the place $P_{t_{0}}$ and
applying the Dedekind-Kummer Theorem [19, Theorem 3.3.7] allows us to write
$f(X)-t_{0}=(X-a)^{4}(X-b)$ for $a,b\in\overline{\mathbb{F}}_{q}$ with
$a\not=b$. ∎

We are now ready to prove that $\operatorname{AGL}_{1}({\mathbb{F}}_{5})$
cannot occur as a geometric monodromy group. The proof will require separate
arguments for even and odd characteristics.

######  Theorem 4.2.

Let $q$ be a prime power with $5\nmid q$ and $f\in{\mathbb{F}}_{q}[X]$ a
polynomial of degree 5. Then
$G(f)\not\cong\operatorname{AGL}_{1}({\mathbb{F}}_{5})$.

###### Proof.

Assume by contradiction that
$G(f)\cong\operatorname{AGL}_{1}({\mathbb{F}}_{5})$. By Lemma 4.1, there are
$t_{0},a,b\in\overline{{\mathbb{F}}}_{q}$ such that $f-t_{0}=(X-a)^{4}(X-b)$.
Since
$\operatorname{Gal}(f(X)-t\mid\overline{{\mathbb{F}}}_{q}(t))\cong\operatorname{Gal}(f(X-c)-(t-d)\mid\overline{{\mathbb{F}}}_{q}(t-d))$
for every $c,d\in\overline{{\mathbb{F}}}_{q}$, we can assume without loss of
generality that $t_{0}=a=0$ and $b\neq 0$.

Odd characteristic. Computing $f^{\prime}(X)=X^{3}(5X-4b)$ shows that
$f^{\prime}(4b/5)=0$, so for $t_{1}=f(4b/5)\in\overline{\mathbb{F}}_{q}$, we
see that $f(X)-t_{1}$ is divisible by $(X-4b/5)^{2}$. Furthermore, since
$X=4b/5$ is not a root of $f^{\prime\prime}(X)=4X^{2}(5X-3b)$, it follows that
$X=4b/5$ is precisely a double root of $f(X)-t_{1}$. Notice that $f(4b/5)\neq
0=f(4b/5)-t_{1}$ and hence $t_{1}\neq 0$, so $X=0$ is not a root of
$f(X)-t_{1}$. This implies that the only repeated root of $f(X)-t_{1}$ is
$X=4b/5$ since the only roots of $f^{\prime}(X)$ are $X=0$ and $X=4b/5$. In
other words, we can write $f(X)-t_{1}=(X-4b/5)^{2}(X-x_{1})(X-x_{2})(X-x_{3})$
for pairwise distinct elements
$4b/5,x_{1},x_{2},x_{3}\in\overline{\mathbb{F}}_{q}$. Finally, let $R$ be any
place of $M$ lying over $P_{t_{1}}$. Then the previous factorization shows
that there is a transposition in $D(R|P_{t_{1}})\leq G(f)$. But $G(f)$
contains no transpositions, and we have a contradiction.

Even characteristic. From now on, we let $x$ be a root of $f(X)-t$ in $M$ and
$f\coloneqq\overline{{\mathbb{F}}}_{q}(x)$. Fix a place $R$ of $M$ lying above
the place $P_{0}$ of $\overline{{\mathbb{F}}}_{q}(t)$. The natural action of
$C_{4}\cong D(R|P_{0})\subseteq G$ on the set of roots of $f(X)-t$ yields
orbits of sizes $4$ and $1$, so there must be two places $Q_{0}$ and $Q_{1}$
of $F$ lying over $P_{0}$ with ramification indices $e(Q_{0}|P_{0})=4$ and
$e(Q_{1}|P_{0})=1$ by [3, Lemma 2.1]. Let $R_{0}$ be a place of $M$ lying over
$Q_{0}$. We have just seen that
$4=|D(R_{0}|P_{0})|=e(R_{0}|P_{0})=e(R_{0}|Q_{0})\cdot e(Q_{0}|P_{0})$, so it
follows that $e(R_{0}|Q_{0})=1$.

Before proceeding, we introduce the following notation: given a function field
$K$ and a place $P$ of $K$, we will write $\hat{K}_{P}$ to denote the
completion of $K$ at $P$ with respect to the $P$-adic metric. In particular,
$\hat{F}_{Q_{0}}=\overline{\mathbb{F}}_{q}((x))$ and
$\hat{\overline{\mathbb{F}}}_{q}(t)_{P_{0}}=\overline{\mathbb{F}}_{q}((t))$.
Using a well-known number theoretical fact (see for example [17, Proposition
II.9.6]), we have
$\textup{Gal}(\hat{M}_{R_{0}}\mid\overline{\mathbb{F}}_{q}((t)))\cong
D(R_{0}|P_{0})\cong C_{4}$. Observe that
$\hat{M}_{R_{0}}\supseteq\hat{F}_{Q_{0}}\supseteq\overline{\mathbb{F}}_{q}((t))$,
so because $[\hat{F}_{Q_{0}}:\overline{\mathbb{F}}_{q}((t))]=e(Q_{0}|P)=4$ and
$[\hat{M}_{R_{0}}:\overline{\mathbb{F}}_{q}((t))]=e(R_{0}|P_{0})=4$ we have
$\hat{M}_{R_{0}}=\hat{F}_{Q_{0}}$. In particular,
$\hat{F}_{Q_{0}}/\overline{\mathbb{F}}_{q}((t))$ is a Galois extension.
Denoting the local Galois group
$\textup{Gal}(\hat{F}_{Q_{0}}\mid\overline{\mathbb{F}}_{q}((t)))$ by
$\hat{G}$, we have $\hat{G}\cong C_{4}$.

The above shows that every root of $f(X)-t$ in $\hat{M}_{R_{0}}$ can be
expressed as an element of $\hat{F}_{Q_{0}}=\overline{\mathbb{F}}_{q}((x))$,
that is, as a Laurent series in $x$. We proceed by showing that if $z\not=x$
is any other root of $f(X)-t$, then we can write $z=x+ux^{i}$ for some $i\geq
2$ and $u\in\overline{\mathbb{F}}_{q}[[x]]$. First, recall that
$f(X)-t=X^{4}(X-b)-t$ so that $b$ is a simple root of $f(X)$. Then by Hensel’s
lifting lemma there is some $\bar{b}\in\overline{\mathbb{F}}_{q}((t))$ such
that we can write $f(X)-t=\bar{f}(X)(X-\bar{b})$ over
$\overline{\mathbb{F}}_{q}((t))$, where
$\bar{f}(X)\in\overline{\mathbb{F}}_{q}((t))[X]$ and $\deg\bar{f}=4$. Further,
the polynomial $\bar{f}$ must be irreducible over
$\overline{\mathbb{F}}_{q}((t))$ since otherwise we could write
$\bar{f}(X)=\bar{g}(X)(X-r)$ for an irreducible
$\bar{g}\in\overline{\mathbb{F}}_{q}((t))[X]$ and some
$r\in\overline{\mathbb{F}}_{q}((t))$, or we could write
$\bar{f}(X)=\bar{h}_{1}(X)\bar{h}_{2}(X)$ for two irreducible quadratic
polynomials $\bar{h}_{1},\bar{h}_{2}\in\overline{\mathbb{F}}_{q}((t))[X]$
having distinct roots in $F_{Q_{0}}$ (since $f-t$ is separable). The former
factorization implies $|\hat{G}|=4\mid 3!$, a clear contradiction, so assume
the latter factorization holds and let $\hat{H}_{1}$ and $\hat{H}_{2}$ be the
splitting fields of $\bar{h}_{1}(X)$ and $\bar{h}_{2}(X)$, respectively, in
$F_{Q_{0}}$. Then $F_{Q_{0}}=H_{1}H_{2}$ so that
$C_{4}\cong\hat{G}=\textup{Gal}(H_{1}H_{2}\mid\overline{\mathbb{F}}_{q}((t)))\subseteq\textup{Gal}(H_{1}\mid\overline{\mathbb{F}}_{q}((t)))\times\textup{Gal}(H_{2}\mid\overline{\mathbb{F}}_{q}((t)))\cong
C_{2}\times C_{2}$, another contradiction. Thus we conclude that $\hat{G}$ is
the Galois group of $\bar{f}(X)$ over $\overline{\mathbb{F}}_{q}((t))$; in
particular, we have that $\hat{G}$ acts transitively on the roots of
$\bar{f}(X)$ in $F_{Q_{0}}$ and hence there is some automorphism
$\tau\in\hat{G}$ of $M$ satisfying $\tau(x)=z$.

Observe that $\langle x\rangle$ is the unique maximal ideal of the ring
$\overline{\mathbb{F}}_{q}[[x]]$, so since $\tau$ is an automorphism of
$\overline{\mathbb{F}}_{q}((x))$ (and hence $\tau$ preserves maximal ideals)
we must have $\langle\tau(x)\rangle=\tau\left(\langle x\rangle\right)=\langle
x\rangle$. Then $z\equiv 0\mod{\langle x\rangle}$ if and only if
$\tau(x)\equiv 0\mod{\langle\tau(x)\rangle}$, and the latter clearly holds.
This allows us to write $z=cx+ux^{i}$ for some $c\in\overline{\mathbb{F}}_{q}$
and for some $i\geq 1$, and we can assume further that $i\geq 2$ since
otherwise we could replace $cx$ by $c^{\prime}x$ for an appropriate
$c^{\prime}\in\overline{\mathbb{F}}_{q}^{*}$ so that this holds. Now computing
$\tau^{4}(x)$ by using $\tau(x)=cx+ux^{i}$ and comparing coefficients with
$\tau^{4}(x)=x$ yields $c^{4}=1$ and hence $c=1$ as we are working over a
field with characteristic $2$. Thus we can write $z=x+ux^{i}$ for some $i\geq
2$ and some $u\in\overline{{\mathbb{F}}}_{q}[[x]]^{*}$.

Observe the following:

$\displaystyle f(z)-t$ $\displaystyle=z^{4}(z-b)-t$
$\displaystyle=\left(x+ux^{i}\right)^{4}(x+ux^{i}-b)-t$
$\displaystyle=\left(x^{4}+u^{4}x^{4i}\right)\left(x-b+ux^{i}\right)-t$
$\displaystyle=x^{4}(x-b)-t+ux^{4+i}+u^{4}x^{4i}(x-b)+u^{5}x^{5i}$
$\displaystyle=ux^{4+i}+u^{4}x^{4i}(x-b)+u^{5}x^{5i},$

where the last equality holds since $x$ is a root of $f(X)-t$. Let
$A=ux^{4+i}+u^{4}x^{4i}(x-b)+u^{5}x^{5i}=f(z)-t$. Since $z$ was chosen to be
another root of $f(X)-t$, we must have $A=0$. But
$\upsilon_{Q_{0}}(A)=\upsilon_{Q_{0}}(x)\cdot\min\limits_{i\geq
2}\\{4+i,4i,5i\\}=4+i\not=\infty=\upsilon_{Q_{0}}(0)$, a contradiction since
$u\not=0$. Thus our initial assumption was false, so
$G(f)\not\cong\textup{AGL}_{1}(\mathbb{F}_{5})$. ∎

##  5\. Degree $5$: if $D_{5}$ or $A_{5}$ occurs, then $5\mid q^{2}-1$

Assume $q$ is a prime power and $f\in{\mathbb{F}}_{q}[X]$ is a separable
polynomial of degree $5$. Let $t$ be transcendental over ${\mathbb{F}}_{q}$
and let $M$ be the splitting field of $f(X)-t$. Let $A(f)$ and $G(f)$ be the
arithmetic and geometric monodromy groups of $f$, respectively.

######  Theorem 5.1.

Suppose that $5\nmid q$ and $A(f)\leq A_{5}$. Then $5\mid q^{2}-1$. In
particular, if $A(f)\leq D_{5}$, then $q^{2}-1$.

###### Proof.

The second assertion follows immediately from the fact that the transitive
copy of $D_{5}$ inside $S_{5}$ lies inside $A_{5}$.

If $q$ is odd, just use the fact that $A(f)\leq A_{5}$ if and only if the
discriminant of $f(X)-t$ is a square in ${\mathbb{F}}_{q}(t)$. When $f(X)$ is
monic of degree 5, the discriminant has the form
$5^{5}t^{4}+\sum_{i=0}^{3}a_{i}t^{i}$. Hence $5$ needs to be a square in
${\mathbb{F}}_{q}$; this implies that either $q$ is an even power of a prime
$p$ or, by quadratic reciprocity, that $q\equiv\pm 1\bmod 5$. In any case,
$5\mid q^{2}-1$.

If $q=2^{n}$ for some $n\geq 1$, one needs to use the characteristic two
analogue of the discrimininant, called _Berlekamp discriminant_ (see [6]). If
$k$ is a field of characteristic 2 and $g\in k[x]$ has degree $n$, the
Berlekamp discriminant of $g$ is an element $\Delta\in k$, that can be
effectively computed using the coefficients of $g$, that has the property that
$\operatorname{Gal}(g)\leq A_{n}$ if and only if the polynomial
$X^{2}+X+\Delta$ has a root in $k$.

Now let $f=X^{5}+aX^{4}+bX^{3}+cX^{2}+dX\in{\mathbb{F}}_{2^{n}}[X]$. We will
show that if $A(f)\leq A_{5}$ then $2\mid n$, and consequently $5\mid q^{2}-1$
once again. One can compute the Berlekamp discriminant $\Delta_{f}$ of $f-t$,
seen as a polynomial over ${\mathbb{F}}_{2^{n}}(t)$, and see that this is
given by an expression of the form $r(t)/s(t)^{2}$, where
$r,s\in{\mathbb{F}}_{2^{n}}(t)$ are two monic polynomials with $\deg r=4$ and
$\deg s=2$. Suppose that $A(f)\leq A_{5}$, and hence $X^{2}+X+\Delta_{f}$ has
a root in ${\mathbb{F}}_{2^{n}}(t)$. Then there are coprime polynomials
$u(t),v(t)\in{\mathbb{F}}_{2^{n}}[t]$, with $v(t)$ monic, such that
$\Delta_{f}=(u(t)^{2}+u(t)v(t))/v(t)^{2}$. Therefore if $r,s$ share a common
factor, this can only have degree $2$ or $4$, and if it has degree $2$ then it
must be of the form $(t+\alpha)^{2}$ for some $\alpha$. Clearly if they share
a factor of degree $4$ then $\Delta_{f}=r(t)/s(t)^{2}=1$, so that $X^{2}+X+1$
has a root in ${\mathbb{F}}_{2^{n}}(t)$ and consequently $2\mid n$. Otherwise
looking at degrees one sees that it must be $\deg u=\deg v$ and
$\deg(u^{2}+uv)=2\deg v$. If the leading coefficient of $u$ is
$\delta\in{\mathbb{F}}_{2^{n}}$, these two conditions, together with the fact
that $r(t)$ is monic of degree $4$, imply that $\delta^{2}+\delta+1=0$ and
consequently that $2\mid n$. ∎

###### Proof of Theorem 2.3.

First, suppose that $5\mid q(q-1)$. Then by Lemmas 3.1 and 3.2, $G_{5}(q)=5$.

Now suppose that $5\nmid q(q-1)$. Then by Lemma 3.2 we have $G_{5}(q)>5$. If
$5\mid q+1$, it is known (see for example [7, Section 3]) that degree $5$
Dickson polynomials of the first kind, e.g. $f=X^{5}-5X^{3}+5X$, satisfy
$G(f)=A(f)=D_{5}$. Hence $G_{5}(q)=10$.

Finally, suppose that $5\nmid q(q^{2}-1)$. Then by Lemma 3.2 and Theorems 4.2
and 5.1 we cannot have $G_{5}(q)=5,10,20$ or $60$. Hence $G_{5}(q)=120$. ∎

######  Remark 5.2.

Notice that the $q/10$ asymptotic when $5|q+1$ was in fact obtained in [14]
using Dickson Polynomials and an independent approach.

##  6\. Computational examples

Let us show with a couple of explicit examples how the number of totally split
places compares to the theoretical estimate given by Proposition 2.2. We pick
examples with $G_{5}(q)=120$; for each of these values of $q$ we pick
polynomials $f$ with $G(f)=A(f)\cong S_{5}$ for $5\nmid q(q^{2}-1)$. As proved
in Theorem 2.3, it is not possible to do better for these $q$’s.

In order to construct polynomials with geometric monodromy (and therefore also
arithmetic monodromy) $S_{5}$, one can use the following well-known group
theoretical fact (see [11]): if $G\leq S_{5}$ is a transitive subgroup
containing a transposition and a cycle of prime length $\ell>2$, then
$G=S_{5}$. In order to force the geometric monodromy group to contain two such
elements it is enough, by ramification arguments, to pick
$g(X)\in{\mathbb{F}}_{q}[X]$ irreducible of degree 3, and set $f=X^{2}g(X)$.

$\qquad q\qquad$ | $\mathcal{T}^{1}_{split}(f)$ | $\lfloor\frac{q}{120}\rfloor$
---|---|---
$2^{13}$ | 78 | 68
$2^{15}$ | 278 | 273
$2^{17}$ | 1088 | 1092
$2^{19}$ | 4332 | 4369
(a) $f=X^{2}(X^{3}+X+1)$

$\qquad q\qquad$ | $\mathcal{T}^{1}_{split}(f)$ | $\lfloor\frac{q}{120}\rfloor$
---|---|---
$3^{7}$ | 21 | 18
$3^{9}$ | 159 | 164
$3^{11}$ | 1474 | 1476
$3^{13}$ | 13338 | 13286
(b) $f=X^{2}(X^{3}-X+1)$

$\qquad q\qquad$ | $\mathcal{T}^{1}_{split}(f)$ | $\lfloor\frac{q}{120}\rfloor$
---|---|---
$19583$ | 156 | 163
19597 | 163 | 163
19687 | 155 | 164
19753 | 194 | 164
19793 | 179 | 164
19913 | 189 | 165
19927 | 160 | 166
19963 | 162 | 166
19993 | 156 | 166
19997 | 161 | 166
(c) singlelinecheck=true

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# An Optimal Joint Antenna and User Selection Algorithm for QoS-based CR-NOMA

Ömer Faruk Akyol, Fethi Okta, Semiha Tedik Başaran Ö. F. Akyol, F. Okta and S.
Tedik Başaran are with the Department of Electronics, and Communications
Engineering, Istanbul Technical University, 34467 Istanbul, Turkey, e-mails:
{akyolo15, okta15<EMAIL_ADDRESS>and Ö. F. Akyol is also with HİSAR
Laboratory, Informatics and Information Security Research Center (BİLGEM),
TÜBİTAK, 41470 Kocaeli, Turkey.

###### Abstract

Both non-orthogonal multiple access (NOMA), which can serve multiple users
simultaneously and on the same frequency, and cognitive radio (CR) can
contribute to eliminating the spectrum scarcity problem. In this work, an
uplink CR-based NOMA (CR-NOMA) system, which is equipped with multiple users
and a base station with a multi-antenna, is proposed to improve spectral
efficiency. By considering the users’ quality of service (QoS), the system
performance of successive interference cancellation (SIC) is investigated in
this system. Two different antenna and secondary user selection algorithms are
proposed to improve the outage performance and retard the effect of the error
floor. The multi-antenna CR-NOMA with QoS-based SIC system outperforms
conventional channel state information (CSI)-based SIC. In addition, it is
shown that the outage performance of this system on both proposed algorithms
is better than in the case of not using the algorithm. The closed-form outage
probability expression of this system for the suboptimal joint antenna and
user selection algorithm is derived. Furthermore, when the proposed algorithms
are not used, the closed-form expression of the outage probability for this
system with a single-antenna base station is derived. Extensive simulation
results verify the accuracy of theoretical analyses.

###### Index Terms:

Non-orthogonal multiple access (NOMA), Cognitive radio (CR), Multi-Antenna,
Quality of Service (QoS), Channel State Information (CSI)

##  I Introduction

In wireless communication, limited and scarce radio spectrum is an important
problem, especially for the sub-6GHz spectrum. In conventional orthogonal
multiple access (OMA), increasing the number of users limits orthogonal
resources and the spectrum. On the other hand, non-orthogonal multiple access
(NOMA) can serve multiple users efficiently by sharing the same spectrum
instead of limiting the spectrum by assigning each of these users to different
bands of the spectrum[1]. Due to its ability to simultaneously serve several
users while achieving great spectrum efficiency, and system capacity, and
providing low complexity and a fairness/throughput tradeoff, NOMA is seen as a
promising technology for the future generation of mobile communication
networks [2]. In addition, the cognitive radio (CR) concept is another
promising technique to solve this radio spectrum scarcity problem. In CR,
there are two different types of users, which are namely the primary user and
the secondary user [3]. The primary user is privileged as the secondary user.
Hence, the secondary user can use the spectrum only when the primary user is
in the idle state [4].

To increase the efficiency of spectrum usage, NOMA can be combined with CR
[5]. In this way, CR-NOMA has an important potential to provide the growing
requirements of users effectively in wireless communication [6, 7]. The
authors in [8] examined the outage probability of an overlay CR-based NOMA
network. Also, the downlink NOMA networks combined CR over the underlay
paradigm has been investigated in [9]. The authors in [10] focused on a
downlink CR-based NOMA system including multiple secondary users and multiple
relays to enhance the outage performance. In addition, by utilizing multiple
secondary users as the relay in a downlink CR-based NOMA network, the study in
[11] presents cooperation to CR-based NOMA. In [12], an uplink CR-based NOMA
system including a rate splitting process, which provides to allocate the
transmit power efficiently by the secondary user for the highest achievable
rate, has been proposed. In [13], an overlay CR-NOMA network was analysed in
terms of the physical layer security and the secrecy performance of primary
users was aimed to improve.

On the other hand, in NOMA networks, serving multiple users at the same time
and frequency causes multiple-access interference. To eliminate this
interference successfully, successive interference cancellation (SIC) is
utilized [14]. The decoding order of the SIC is an important issue for the
performance and applicability of the system. In the literature, there are two
important SIC decoding orders based on the quality of service (QoS) and the
channel state information (CSI) [15],[16]. According to the QoS criterion, the
signal of the primary user, who has higher priority QoS requirements, is
decoded first. Subsequent stages of the SIC decode the signals of the other
users. On the other hand, based on the CSI criteria, the signal of the users
who have a better CSI than the others is first decoded, and then the remaining
users are decoded using SIC following the CSI state [17]. The outage
performance results of selecting the SIC decoding orders based on both CSI and
QoS in uplink NOMA systems with multi-user are investigated [18],[19] and the
outage performances of both SIC methods in downlink NOMA systems are presented
in [20].

However, there is no study focusing on the subject of SIC decoding orders
based on either QoS or CSI for the CR-NOMA networks with multiple antennas at
the receiver side. In this paper, we consider an uplink CR-NOMA system with a
multi-antenna base station and multi-user. The effect of SIC decoding order on
outage performance will be investigated in this proposed CR-NOMA system by
comparing QoS-based SIC with CSI-based SIC. Moreover, we will select the
optimal secondary user and receiver antenna to improve the outage performance
of the system model through the algorithms that will be proposed in the
following parts of this paper. Hence, we achieve a surprising improvement in
the outage performance of the proposed NOMA system with QoS-based SIC compared
with CSI-based SIC. The main contributions of this work are given as follows:

  * •

An uplink CR-NOMA network with multiple secondary users and multi-antenna is
proposed. To improve the outage performance of the proposed system with QoS-
based SIC, we propose two different joint antenna and user selection
algorithms.

  * •

In the proposed system, through these proposed algorithms, the QoS-based SIC
is compared to the CSI-based SIC in terms of outage probability. Numerical
results show that the outage performance of the QoS-based SIC outperforms the
CSI-based SIC by using these proposed algorithms. Furthermore, both proposed
algorithm provides a significant gain in terms of the outage performance of
this system by increasing the number of antenna or secondary user.

  * •

The outage probability of QoS-based SIC over the proposed algorithms is
compared in both perfect CSI and imperfect CSI.

  * •

The closed-form expression of the outage probability of the QoS-based NOMA for
the suboptimal joint antenna and user selection algorithm in the proposed
system is derived and then verified with simulation results. Moreover, to show
whether the optimal joint antenna and user selection algorithm works optimally
in this system with QoS-based SIC, the obtained simulation results are
verified by exhaustive analysis.

The rest of the paper is organized as follows. In Section II, the CR-NOMA
system model is presented and this system is briefly investigated over the
CSI-based SIC. In Section III, the same system model is investigated over the
QoS-based SIC in detail and two algorithms are proposed to increase the outage
performance of the QoS-Based SIC in this system. Section IV discusses
numerical and simulation results. Finally, the conclusion is given in Section
V.

##  II Conventional NOMA Network With CSI-based SIC

Figure 1: The considered uplink CR-NOMA system model with multi-antenna.

In this section, we present the proposed multi-user multi-antenna CR-NOMA
system model as shown in Fig.1. In this section, the CSI-based SIC in the
proposed CR-NOMA system is investigated. In the next section, the QoS-based
SIC is investigated for the same NOMA network. In this system model, there are
$M+1$ users and one of these users is the primary user, stated as $D_{0}$, and
the others are secondary users, stated as $D_{m}$ where $m=1,\ldots,M$. The
target data rates of all secondary users and the primary user are $R_{s}^{th}$
and $R_{0}^{th}$, respectively, to detect their signals. The entire mobile
user is equipped with a single antenna and the base station is equipped with
$K$ antenna and $k$ is the antenna index where $k=1,\ldots,K$. The channel
gains of the primary user and $m$th secondary user at the $k$th antenna of the
base station are denoted by $h_{0}^{k}$ and $h_{m}^{k}$, respectively. In
addition, all channels are modelled as Rayleigh fading. The mean values of the
primary user’s channel and the secondary users’ channels are denoted by
$\mathbb{E}\left[\left|h_{0}^{k}\right|^{2}\right]=\Omega_{0}$ and
$\mathbb{E}\left[\left|h_{m}^{k}\right|^{2}\right]=\Omega_{m}$, respectively,
where $\mathbb{E}\left[.\right]$ is the expectation operator. In the proposed
uplink CR-NOMA network, the received signal received at $k$th antenna of the
base station by transmitted the information signals over $D_{0}$ and $D_{m}$
is given by

$\displaystyle y_{S}=$
$\displaystyle\sqrt{P_{S}}h_{0}^{k}x_{PU}+\sqrt{P_{S}}h_{m}^{k}x_{SU}+w_{S},$
(1)

where $x_{PU}$ and $x_{SU}$ are the information signals of $D_{0}$ and
$D_{m}$, respectively, which these signals have a unit power in expectation.
The users’ transmit powers are assumed to be identical and equal to $P_{S}$
and $w_{S}\sim CN\left(0,\sigma^{2}\right)$ is the additive white Gaussian
noise at the base station with mean zero and variance $\sigma^{2}$. The SIC
decoding order can be determined simply by using the users’ channel conditions
in the power-domain NOMA and CR-NOMA networks as in CSI-based SIC [18],[21].
Similar to a power-domain NOMA network serving two users with different
channel conditions simultaneously, if $D_{m}$ is considered strong and $D_{0}$
weak in terms of channel conditions, firstly $D_{m}$ can be decoded at $k$th
antenna of the base station by CSI-based SIC under the effect of interference
by $D_{0}$ based on the following metric:

$\log\left(1+\frac{\bar{\gamma}\left|h_{m}^{k}\right|^{2}}{1+\bar{\gamma}\left|h_{0}^{k}\right|^{2}}\right)\geq
R_{s}^{th}.$ (2)

After that, $D_{0}$ is decoded if $D_{m}$ is decoded successfully after
process in (2). By applying CSI-based SIC process on $k$th antenna, $D_{0}$
and $D_{m}$ ’s achievable data rates are given by, respectively,

$\displaystyle R_{0}^{CSI}=$
$\displaystyle\log\left(1+\bar{\gamma}\left|h_{0}^{k}\right|^{2}\right),$ (3)
$\displaystyle R_{m}^{CSI}=$
$\displaystyle\log\left(1+\frac{\bar{\gamma}\left|h_{m}^{k}\right|^{2}}{1+\bar{\gamma}\left|h_{0}^{k}\right|^{2}}\right),$
(4)

where $\bar{\gamma}=\frac{P_{S}}{\sigma^{2}}$ is the average signal-to-noise
ratio (SNR).

##  III NOMA Network With QoS-based SIC

In CR-NOMA applications, selecting SIC decoding orders is frequently preferred
based on the users’ QoS requirements [22]. In the proposed system based on CR-
NOMA, the QoS requirements of the primary user with poor channel conditions
compared with the secondary users should be guaranteed [23]. However, this
system sacrifices the performance of the secondary users with better channel
conditions to guarantee the primary user’s QoS requirements. Hence, the
primary user’s target data rate should be lower than secondary users i.e.
$R_{0}^{th}<R_{s}^{th}$. Similar to an uplink NOMA network serving two users
simultaneously, one of which is privileged over the other, the primary user
and one of the secondary users simultaneously transmit their messages to the
base station in this system model. In the $k$th antenna of the base station,
as the first stage of QoS-based SIC, firstly $D_{0}$ should be decoded due to
the QoS requirements under the effect of interference by $D_{m}$ based on the
following metric:

$\log\left(1+\frac{\bar{\gamma}\left|h_{0}^{k}\right|^{2}}{1+\bar{\gamma}\left|h_{m}^{k}\right|^{2}}\right)\geq
R_{0}^{th}.$ (5)

If $D_{0}$ is decoded and removed successfully after process in (5), $D_{m}$
can be decoded. During the QoS-based SIC process, $D_{0}$ and $D_{m}$’s
achievable data rates are given by, respectively,

$\displaystyle R_{0}^{QoS}=$
$\displaystyle\log\left(1+\frac{\bar{\gamma}\left|h_{0}^{k}\right|^{2}}{1+\bar{\gamma}\left|h_{m}^{k}\right|^{2}}\right),$
(6) $\displaystyle R_{m}^{QoS}=$
$\displaystyle\log\left(1+\bar{\gamma}\left|h_{m}^{k}\right|^{2}\right).$ (7)

###  III-A Joint Antenna and User Selection Algorithms

In the proposed CR-NOMA system, to improve the outage performance of both QoS-
based SIC and the entire network, user and antenna selection procedures will
be suggested and applied. In the proposed CR-NOMA network with QoS-based SIC,
during the first step of SIC, the primary user’s signal is decoded. After the
SIC process, the secondary user’s signal is decoded. The outage probability of
this proposed network with QoS-based SIC can be given by

$\mathrm{P}^{QoS}=\mathrm{Pr}\left(R_{0}^{QoS}<R_{0}^{th}\right)+\mathrm{Pr}\left(R_{0}^{QoS}>R_{0}^{th},R_{m}^{QoS}<R_{s}^{th}\right),$
(8)

where $\mathrm{Pr}(.)$ is the probability operator. Decoding of the secondary
user’s signal should be guaranteed after the SIC process to improve the entire
system’s outage performance. Hence, both to increase the outage performance of
QoS-based SIC and not restrict decoding the selected secondary user’s signal
at the two-stage of SIC, two algorithms are suggested, namely Algorithm (1)
and Algorithm (2) respectively. In addition, with these proposed Algorithms,
it is aimed to improve both the SIC decoding and the outage performance of the
proposed system model by applying both antenna selection among $K$ antennas
and user selection among $M$ secondary users. The best antenna and secondary
user, which will improve both the performance of the SIC process and the
performance of the entire CR-NOMA network, are selected between multiple
antennas and multiple secondary users based on both algorithms.

####  III-A1 The Suboptimal Joint Antenna and User Selection Algorithm

In Algorithm (1), firstly, before starting the SIC process, the strongest data
rate between the antennas and the secondary users should be compared with the
target data rate of the secondary users, namely $R_{s}^{th}$, to control
whether this network is completely in outage as in Line 1–1. The data rate,
namely $R_{s}^{\max}$, can be given by

$\displaystyle R_{s}^{\max}=\max(\max(R_{m}^{k})),$ (9)

where $R_{m}^{k}=\log_{2}(1+\bar{\gamma}|h_{m}^{k}|^{2})$. If $R_{s}^{\max}$
is bigger than $R_{s}^{th}$, the best antenna and user selection steps can be
applied. However, if not, the outage occurs. In case of no outage, the best
antenna is selected based on the maximum channel gain between the primary user
and all antennas to improve the performance of SIC as in Line 1. Also, to
minimize the performance degradation due to interference occurring by the
secondary user in the process of SIC, which is being applied to decode the
primary user’s signal, the secondary user with the weakest data rate, which is
greater than $R_{s}^{th}$, between the selected antenna and the secondary
users is selected as in Line 1–1. Therefore, after the SIC operation, an
outage in the proposed network due to the data rate of the selected secondary
user being lower than $R_{s}^{th}$ is avoided. The indexes of the selected
antenna and secondary user are given by

$\displaystyle k^{*}=$
$\displaystyle\underset{k=1,\ldots,K}{\operatorname*{argmax}}(|h_{0}^{k}|^{2}),$
(10) $\displaystyle m^{*}=$ $\displaystyle\underset{R_{m}^{k^{*}}\geq
R_{s}^{th}}{\operatorname*{arg\,min}}(|h_{m}^{k^{*}}|^{2}).$ (11)

Hence, the primary user’s data rate is calculated according to the $k^{*}$th
antenna and $m^{*}$th secondary user for the process of SIC in Line 1. Before
the SIC process, the primary user’s data rate is given by

$\displaystyle
R_{0}^{QoS,1}=\log_{2}\Big{(}1+\frac{\bar{\gamma}|h_{0}^{k^{*}}|^{2}}{1+\bar{\gamma}|h_{m^{*}}^{k^{*}}|^{2}}\Big{)}.$
(12)

If $R_{0}^{QoS,1}$ is bigger than $R_{0}^{th}$ under the interference by the
secondary user’s signal, the process of SIC can be carried out successfully
and the outage does not occur. If not, the outage occurs as in Line 1–1. On
the other hand, if the secondary user cannot be selected since no secondary
user’s data rate is greater than $R_{s}^{th}$ between the $k^{*}$th antenna
and all secondary users, i.e. $\max(R_{m}^{k^{*}})<R_{s}^{th}$, provided the
data rate of the secondary user to be selected is greater than $R_{s}^{th}$,
the channel with weakest data rate between the all secondary users and all
antennas are selected. Hence, over this channel, the antenna and the secondary
user are selected as in Line 1–1. After that, in Line 1, the primary user’s
data rate is calculated according to the reselected antenna and secondary user
for the process of SIC. Hence, the primary user’s data rate can be expressed
as

$\displaystyle
R_{0}^{QoS,2}=\log_{2}\Big{(}1+\frac{\bar{\gamma}|h_{0}^{k^{+}}|^{2}}{1+\bar{\gamma}|h_{m^{+}}^{k^{+}}|^{2}}\Big{)}.$
(13)

If $R_{0}^{QoS,2}$ is bigger than $R_{0}^{th}$ under the interference by the
secondary user’s signal, the process of SIC can be carried out successfully
and the outage does not occur. If not, the outage occurs as in Line 1–1.
Moreover, the closed-form expression of the outage probability for Algorithm
(1) is obtained as in (29) through the following procedures. Firstly, to
improve the outage performance of the proposed QoS-based system, the secondary
user’s signal with the weakest data rate bigger than $R_{s}^{th}$ among $M$
secondary users is selected if there is any the secondary user with the data
rate bigger than $R_{s}^{th}$ in this proposed network. In case there is no
secondary user with a data rate bigger than $R_{s}^{th}$, the outage occurs.
Hence, the outage probability of this case can be expressed as

$J_{1}=\mathrm{Pr}\left(R_{s}^{\max}<R_{s}^{th}\right)=F_{Y}(\gamma_{s})^{MK}=\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{MK},$
(14)

where $Y=\bar{\gamma}|h_{m}^{k}|^{2}$. The probability density function (PDF)
and the cumulative distribution function (CDF) of $Y$ are equal to
$f_{Y}(y)=\frac{1}{\tilde{\Omega}_{m}}e^{-\frac{y}{\tilde{\Omega}_{m}}}$ and
$F_{Y}(y)=1-e^{-\frac{y}{\tilde{\Omega}_{m}}}$, respectively. It is assumed
that $\tilde{\Omega}_{0}=\bar{\gamma}\Omega_{0}$ and
$\tilde{\Omega}_{m}=\bar{\gamma}\Omega_{m}$. The threshold SNRs of the primary
user and each of the secondary users are $\gamma_{s}=2^{R_{s}^{th}}-1$ and
$\gamma_{0}=2^{R_{0}^{th}}-1$, respectively. After selecting the antenna with
the strongest channel gain between the primary user and all antennas as in
(10), the probability of not having any secondary users with a data rate
bigger than $R_{s}^{th}$ at the selected $k^{*}$th antenna can be written by

$\displaystyle J_{2}=$
$\displaystyle\mathrm{Pr}\left(\max\left(\log_{2}(1+\bar{\gamma}|h_{m}^{k^{*}}|^{2})\right)<R_{s}^{th}\right)=F_{Y}(\gamma_{s})^{M}.$
(15)

If there is any secondary user with a data rate bigger than $R_{s}^{th}$ at
the selected antenna, in case there are $g$ secondary users with a data rate
lower than $R_{s}^{th}$, the probability can be expressed using the
probability mass function (PMF) as

$\displaystyle\mathrm{Pr}\left(G=g\right)=$
$\displaystyle\binom{M}{g}F_{Y}(\gamma_{s})^{g}\left(1-F_{Y}(\gamma_{s})\right)^{M-g}$
(16) $\displaystyle=$
$\displaystyle\binom{M}{g}\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{g}\left(e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{M-g}.$

It is assumed that there are $M-g$ secondary users with a data rate bigger
than $R_{s}^{th}$ among the selected $k^{*}$th antenna. $M-g$ out of $M$
secondary users will be distributed over the interval
$\left[\gamma_{s},\infty\right)$ and the expression of the PDF for each of
these $M-g$ secondary users is
$f_{\tilde{Y_{1}}}(y)=\frac{1}{\tilde{\Omega}_{m}}e^{-\frac{\left(y-\gamma_{s}\right)}{\tilde{\Omega}_{m}}}$,
where $y\in\left[\gamma_{s},\infty\right)$. Hence, to obtain the expression of
the PDF for the secondary user with the minimum data rate bigger than
$R_{s}^{th}$ among these $M-g$ secondary users, firstly
$\mathrm{Pr}(\tilde{Y_{1}}>y\mid G=g)$ is calculated as follows

$\displaystyle\mathrm{Pr}(\tilde{Y_{1}}>y\mid G=g)=$
$\displaystyle\left(e^{-\frac{\left(y-\gamma_{s}\right)}{\tilde{\Omega}_{m}}}\right)^{M-g}.$
(17)

Then, $\mathrm{Pr}(\tilde{Y_{1}}>y)$ is calculated as

$\displaystyle\mathrm{Pr}(\tilde{Y_{1}}>y)=$
$\displaystyle\mathrm{Pr}(\tilde{Y_{1}}>y\mid G=g)\mathrm{Pr}\left(G=g\right)$
(18) $\displaystyle=$
$\displaystyle\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}+e^{-\frac{y}{\tilde{\Omega}_{m}}}\right)^{M}-\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{M}.$

Hence, the expression of $F_{\tilde{Y_{1}}}(y)$ can be written by

$F_{\tilde{Y_{1}}}(y)=1-\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}+e^{-\frac{y}{\tilde{\Omega}_{m}}}\right)^{M}+\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{M}.$
(19)

Finally, the expression of $f_{\tilde{Y_{1}}}(y)$ is obtained by

$f_{\tilde{Y_{1}}}(y)=\frac{M}{\tilde{\Omega}_{m}}e^{-\frac{y}{\tilde{\Omega}_{m}}}\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}+e^{-\frac{y}{\tilde{\Omega}_{m}}}\right)^{M-1}.$
(20)

Over the selected $k^{*}$th antenna and $m^{*}$th secondary user, if
$R_{0}^{QoS,1}$ is lower than $R_{0}^{th}$, the outage occurs. The closed-form
expression of this outage probability is calculated using
$f_{\tilde{Y_{1}}}(y)$ as

$\displaystyle J_{3}=$
$\displaystyle\mathrm{Pr}\left(\frac{\bar{\gamma}|h_{0}^{k^{*}}|^{2}}{1+\bar{\gamma}|h_{m^{*}}^{k^{*}}|^{2}}<\gamma_{0}\right)=\int_{\gamma_{s}}^{\infty}F_{X}(\gamma_{0}+y\gamma_{0})^{K}f_{\tilde{Y_{1}}}\left(y\right)\mathrm{d}y$
$\displaystyle=$
$\displaystyle\frac{M}{\tilde{\Omega}_{m}}\sum_{a=0}^{K}\sum_{b=0}^{M-1}\left(-1\right)^{a}\binom{K}{a}\binom{M-1}{b}\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{M-1-b}$
$\displaystyle\qquad\qquad\times
e^{-\gamma_{s}\left(\frac{\gamma_{0}}{\tilde{\Omega}_{0}}+\frac{b+1}{\tilde{\Omega}_{m}}\right)}\frac{e^{-\gamma_{0}\frac{a}{\tilde{\Omega}_{0}}}}{\left(\frac{\gamma_{0}}{\tilde{\Omega}_{0}}+\frac{b+1}{\tilde{\Omega}_{m}}\right)},$
(21)

where $X=\bar{\gamma}|h_{0}^{k}|^{2}$. The PDF and CDF of $X$ are equal to
$f_{X}(x)=\frac{1}{\tilde{\Omega}_{0}}e^{-\frac{x}{\tilde{\Omega}_{0}}}$ and
$F_{X}(x)=1-e^{-\frac{x}{\tilde{\Omega}_{0}}}$, respectively. If there is no
secondary user with a data rate bigger than $R_{s}^{th}$ at the selected
$k^{*}$th antenna, provided there is any secondary user with a data rate
bigger than $R_{s}^{th}$ for any antenna, the channels, which can provide a
data rate bigger than $R_{s}^{th}$ between the secondary users and the
antennas, are checked and detected. Among these detected channels, the one
with the lowest data rate is selected. Hence, both the antenna and the
secondary user are selected over the selected channel. The probability of
having $v$ channels, which provide a data rate lower than $R_{s}^{th}$ between
the secondary users and the antennas, can be expressed as

$\displaystyle\mathrm{Pr}\left(V=v\right)=$
$\displaystyle\binom{KM}{v}\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{v}\left(e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{KM-v}.$
(22)

Hence, it is assumed that there are $KM-v$ channels with a data rate bigger
than $R_{s}^{th}$ between the secondary users and the antennas. $KM-v$ out of
$KM$ channels will be distributed over the interval
$\left[\gamma_{s},\infty\right)$ and the expression of the PDF for each of
these $KM-v$ channels is
$f_{\tilde{Y_{2}}}(y)=\frac{1}{\tilde{\Omega}_{m}}e^{-\frac{\left(y-\gamma_{s}\right)}{\tilde{\Omega}_{m}}}$,
where $y\in\left[\gamma_{s},\infty\right)$. To obtain the expression of PDF
for the channel that will provide the minimum data rate bigger than
$R_{s}^{th}$ among $KM-v$ channels, $\mathrm{Pr}(\tilde{Y_{2}}>y\mid V=v)$ is
calculated as follows

$\displaystyle\mathrm{Pr}(\tilde{Y_{2}}>y\mid V=v)=$
$\displaystyle\left(e^{-\frac{\left(y-\gamma_{s}\right)}{\tilde{\Omega}_{m}}}\right)^{KM-v}.$
(23)

Hence, $\mathrm{Pr}(\tilde{Y_{2}}>y)$ is calculated using
$\mathrm{Pr}(\tilde{Y_{2}}>y\mid V=v)$ and $\mathrm{Pr}\left(V=v\right)$ as

$\displaystyle\mathrm{Pr}(\tilde{Y_{2}}>y)=$
$\displaystyle\mathrm{Pr}(\tilde{Y_{2}}>y\mid V=v)\mathrm{Pr}\left(V=v\right)$
(24) $\displaystyle=$
$\displaystyle\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}+e^{-\frac{y}{\tilde{\Omega}_{m}}}\right)^{KM}-\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{KM}.$

The CDF expression of $F_{\tilde{Y_{2}}}(y)$ can be given by

$F_{\tilde{Y_{2}}}(y)=1-\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}+e^{-\frac{y}{\tilde{\Omega}_{m}}}\right)^{KM}+\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{KM}.$
(25)

Finally, the PDF expression of $f_{\tilde{Y_{2}}}(y)$ is obtained by

$f_{\tilde{Y_{2}}}(y)=\frac{KM}{\tilde{\Omega}_{m}}e^{-\frac{y}{\tilde{\Omega}_{m}}}\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}+e^{-\frac{y}{\tilde{\Omega}_{m}}}\right)^{KM-1}$
(26)

Over the selected $k^{+}$th antenna and $m^{+}$th secondary user, if
$R_{0}^{QoS,2}$ is lower than $R_{0}^{th}$, the outage occurs. The closed-form
expression of this outage probability is calculated using
$f_{\tilde{Y_{2}}}(y)$ as

$\displaystyle J_{4}=$
$\displaystyle\mathrm{Pr}\left(\frac{\bar{\gamma}|h_{0}^{k^{+}}|^{2}}{1+\bar{\gamma}|h_{m^{+}}^{k^{+}}|^{2}}<\gamma_{0}\right)=\int_{\gamma_{s}}^{\infty}F_{X}(\gamma_{0}+y\gamma_{0})f_{\tilde{Y_{2}}}\left(y\right)\mathrm{d}y$
(27) $\displaystyle=$
$\displaystyle\frac{KM}{\tilde{\Omega}_{m}}\sum_{c=0}^{1}\sum_{d=0}^{KM-1}\left(-1\right)^{c}\binom{KM-1}{d}e^{-\gamma_{s}\left(\frac{\gamma_{0}}{\tilde{\Omega}_{0}}+\frac{d+1}{\tilde{\Omega}_{m}}\right)}$
$\displaystyle\qquad\qquad\times\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{KM-1-d}\frac{e^{-\gamma_{0}\frac{c}{\tilde{\Omega}_{0}}}}{\left(\frac{\gamma_{0}}{\tilde{\Omega}_{0}}+\frac{d+1}{\tilde{\Omega}_{m}}\right)}.$

Finally, the closed-form expression of the outage probability for Algorithm
(1) can be written as

$\mathrm{P}^{QoS}=J_{1}+\left(1-J_{1}\right)\left[\left(1-J_{2}\right)J_{3}+J_{2}J_{4}\right].$
(28)

Using (28), the closed-form theoretical expression of Algorithm (1) is given
in (29). On the other hand, in case the base station with a single antenna,
the secondary user with minimum channel gain among $M$ secondary user is
selected to improve the outage performance of the NOMA network with QoS-based
SIC as in [18]. Therefore, the outage performances of Algorithm (1) and
secondary user selection method for QoS-based SIC in [18] are investigated in
the numerical results section in the case of a single antenna base station.

$\displaystyle\mathrm{P}^{QoS}=\left(1-\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{MK}\right)\Bigg{[}\left(1-\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{M}\right)\frac{M}{\tilde{\Omega}_{m}}\sum_{a=0}^{K}\sum_{b=0}^{M-1}\left(-1\right)^{a}\binom{K}{a}\binom{M-1}{b}\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{M-1-b}\frac{e^{-\gamma_{0}\frac{a}{\tilde{\Omega}_{0}}}}{\left(\frac{\gamma_{0}}{\tilde{\Omega}_{0}}+\frac{b+1}{\tilde{\Omega}_{m}}\right)}e^{-\gamma_{s}\left(\frac{\gamma_{0}}{\tilde{\Omega}_{0}}+\frac{b+1}{\tilde{\Omega}_{m}}\right)}$
(29)
$\displaystyle+\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{M}\frac{KM}{\tilde{\Omega}_{m}}\sum_{c=0}^{1}\sum_{d=0}^{KM-1}\left(-1\right)^{c}\binom{KM-1}{d}e^{-\gamma_{s}\left(\frac{\gamma_{0}}{\tilde{\Omega}_{0}}+\frac{d+1}{\tilde{\Omega}_{m}}\right)}\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{KM-1-d}\frac{e^{-\gamma_{0}\frac{c}{\tilde{\Omega}_{0}}}}{\left(\frac{\gamma_{0}}{\tilde{\Omega}_{0}}+\frac{d+1}{\tilde{\Omega}_{m}}\right)}\Bigg{]}+\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{m}}}\right)^{MK}$

1

2

input : $K$, $M$, $\mathcal{G}[M,K-1]$, $\mathcal{L}[1...K]$

output : $R^{QoS}$, $error$

3

4if  _$R_{s}^{\max} <R_{s}^{th}$_ then

5 $error$;

6

7else

8 $k^{*}\leftarrow\underset{k=1,\ldots,K}{\mathrm{argmax}}(|h_{0}^{k}|^{2})$

9 if  _$\max(R_{m}^{k^{*}})\geq R_{s}^{th}$_ then

10

11 for  _$m\leftarrow 1$ to $M$_ do

12 $\mathcal{L}[m]\leftarrow 0$

13 if  _$R_{m}^{k^{*}}\geq R_{s}^{th}$_ then

14 $\mathcal{L}[m]\leftarrow|h_{m}^{k^{*}}|^{2}$

15 end if

16

17 end for

18

19
$m^{*}\leftarrow\underset{m}{\mathrm{argmin}}(\mathcal{L}|\mathcal{L}[m]>0)$

20
$R^{QoS}=\log_{2}\Big{(}1+\frac{\bar{\gamma}|h_{0}^{k^{*}}|^{2}}{1+\bar{\gamma}|h_{m^{*}}^{k^{*}}|^{2}}\Big{)}$

21 else

22 for  _$m\leftarrow 1$ to $M$_ do

23

24 for  _$k\leftarrow 1$ to $K$ / $k^{*}$_ do

25 $\mathcal{G}[m,k]\leftarrow 0$

26 if  _$R_{m}^{k}\geq R_{s}^{th}$_ then

27 $\mathcal{G}[m,k]\leftarrow|h_{m}^{k}|^{2}$

28 end if

29

30 end for

31

32 end for

33

34
$m^{+},k^{+}\leftarrow\underset{m}{\mathrm{argmin}}\underset{k}{\mathrm{argmin}}(\mathcal{G}|\mathcal{G}[m,k]>0)$

35
$R^{QoS}=\log_{2}\Big{(}1+\frac{\bar{\gamma}|h_{0}^{k^{+}}|^{2}}{1+\bar{\gamma}|h_{m^{+}}^{k^{+}}|^{2}}\Big{)}$

36 end if

37 if  _$R^{QoS} <R_{0}^{th}$_ then

38 $error$;

39

40 end if

41

42 end if

43

Algorithm 1 The Suboptimal Joint Antenna and User Selection Algorithm

####  III-A2 The Optimal Joint Antenna and User Selection Algorithm

In another proposed algorithm for the CR-NOMA network with QoS-based SIC,
firstly, as at the beginning of the Algorithm (1), to control if this network
is in the outage completely before the SIC process, it should be checked
whether $R_{s}^{\max}$ is bigger than $R_{s}^{th}$. If $R_{s}^{\max}$ is
bigger than $R_{s}^{th}$, the antenna and user selection stages can be
applied. If not, it is accepted that the outage occurs in this network before
the process of SIC as in Line 2–2. In the antenna and user selection stages,
all channels between secondary users and antennas, satisfying the condition
that the data rate is bigger than $R_{s}^{th}$, are detected in Line 2–2.
After that, all possible antennas, which can be used to select the best
antenna, are determined by controlling whether there is any secondary user
with a data rate greater than $R_{s}^{th}$ between each antenna and secondary
users as in Line 2. The secondary users with a data rate greater than
$R_{s}^{th}$ but the lowest data rate among $M$ secondary users is detected
for each of these possible antennas separately as in Line 2. Using the
secondary users detected separately for each of these possible antennas, as
many as the number of all possible antennas that can be selected, the
expression of $|h_{0}^{k}|^{2}/|h_{m^{\star}}^{k}|^{2}$ is repeatedly
calculated and recorded in Line 2. By using these recorded values, the best
antenna and accordingly the best secondary user are selected as in Line 2.
After the stage of antenna and secondary user selection, the process of SIC
will be applied. The primary user’s data rate is given by

$\displaystyle
R_{0}^{QoS,3}=\log_{2}\Big{(}1+\frac{\bar{\gamma}|h_{0}^{k^{\star}}|^{2}}{1+\bar{\gamma}|h_{m^{\star}}^{k^{\star}}|^{2}}\Big{)}.$
(30)

If $R_{0}^{QoS,3}$, which is calculated in Line 2, is greater than
$R_{0}^{th}$, the process of SIC is carried out successfully by being decoded
the primary user’s signal. If not, the outage occurs in the proposed NOMA
network as in Line 2–2.

1

2

input : $K$, $M$, $\mathcal{L}[1...M]$, $\mathcal{H}[1...K]$

output : $R^{QoS}$, $error$

3

4if  _$R_{s}^{\max} <R_{s}^{th}$_ then

5 $error$;

6

7else

8

9 for  _$k\leftarrow 1$ to $K$_ do

10 $Sum\leftarrow 0$

11 $\mathcal{H}[k]\leftarrow 0$

12 for  _$m\leftarrow 1$ to $M$_ do

13 $\mathcal{L}[m]\leftarrow 0$

14 if  _$R_{m}^{k}\geq R_{s}^{th}$_ then

15 $\mathcal{L}[m]\leftarrow|h_{m}^{k}|^{2}$

16 $Sum\leftarrow Sum+1$

17 else

18 $Sum\leftarrow Sum+0$

19 end if

20

21 end if

22

23 end for

24 if  _$Sum >0$_ then

25 $m^{\star}\leftarrow\underset{m}{\arg\min}(\mathcal{L}|\mathcal{L}[m]>0)$

26 $\mathcal{H}[k]\leftarrow|h_{0}^{k}|^{2}/|h_{m^{\star}}^{k}|^{2}$

27 end if

28

29 end for

30 $k^{\star}\leftarrow\underset{k}{\arg\max}(\mathcal{H})$

31
$R^{QoS}\leftarrow\log_{2}\Big{(}1+\frac{\bar{\gamma}|h_{0}^{k^{\star}}|^{2}}{1+\bar{\gamma}|h_{m^{\star}}^{k^{\star}}|^{2}}\Big{)}$

32 if  _$R^{QoS} <R_{0}^{th}$_ then

33 $error$;

34

35 end if

36

37 end if

Algorithm 2 The Optimal Joint Antenna and User Selection Algorithm

###  III-B QoS-based SIC with imperfect channel estimation

In this subsection, we present the channel model that will be used to examine
how well the suggested network with imperfect CSI performs in channels with
independent and identically distributed (i.d.d.) Rayleigh fading. Assuming the
channel estimation is imperfect, the primary pilot symbol assisted channel
estimate $\hat{h}_{0}^{k}$, which is between the primary user and $k$th
antenna, and the secondary pilot symbol assisted channel estimate
$\hat{h}_{m}^{k}$, which is between the $m$th secondary user and $k$th
antenna, differ from true channels $h_{0}^{k}$ and $h_{m}^{k}$ by an
independent complex Gaussian error $\Delta h_{0}^{k}$ and $\Delta h_{m}^{k}$,
respectively, which are with zero-mean and variance $\sigma_{e}^{2}$. Hence,
$\hat{h}_{0}^{k}$ and $\hat{h}_{m}^{k}$ are equal to $h_{0}^{k}+\Delta
h_{0}^{k}$ and $h_{m}^{k}+\Delta h_{m}^{k}$, respectively [24]. Assuming
$\Omega_{0}=\Omega_{m}=1$, the estimate channels $\hat{h}_{0}^{k}$ and
$\hat{h}_{m}^{k}$ are i.i.d. complex Gaussian random variables with zero-mean
and variance $1+\sigma_{e}^{2}$. The true channel gains $h_{0}^{k}$ and
$h_{m}^{k}$ can be written in terms of $\hat{h}_{0}^{k}$ and
$\hat{h}_{m}^{k}$, respectively [25],

$\displaystyle h_{0}^{k}$
$\displaystyle=\Gamma\hat{h}_{0}^{k}+\tilde{h}_{0}^{k}$ (31) $\displaystyle
h_{m}^{k}$ $\displaystyle=\Gamma\hat{h}_{m}^{k}+\tilde{h}_{m}^{k}$

where $\Gamma=\frac{1}{1+\sigma_{e}^{2}}$. Also, $\tilde{h}_{0}^{k}$ and
$\tilde{h}_{m}^{k}$ are i.i.d. complex Gaussian random variables with zero-
mean and variance
$\tilde{\sigma}^{2}=\frac{\sigma_{e}^{2}}{1+\sigma_{e}^{2}}$.

##  IV Numerical And Simulation Results

In this section, for the proposed uplink CR-NOMA scenario, the QoS-based SIC
is compared with the CSI-based SIC in terms of outage probability over two
different algorithm methods and then, in case both perfect channel state and
imperfect channel state, the outage performance of the QoS-based SIC over
these algorithms is investigated according to the different number of antennas
and secondary users. In addition, the outage performance of the QoS-based SIC
for Algorithm (1) is evaluated with numerical results obtained by Monte Carlo
simulations and these numerical results are verified by theoretical results.
Also, the outage performance of the QoS-based SIC for Algorithm (2) is
confirmed by the exhaustive analysis that is a primary method to find the
optimal scheduled secondary user and antenna [26]. On the other hand, to
present the performance of the CSI-based SIC for this proposed system in Fig.
2 and Fig. 6, the antenna with the weakest channel gain between the primary
user and antennas is selected to increase the data rate of the secondary user
as in (4) and, then the secondary user with the strongest channel gain between
the selected antenna and secondary users is selected.

Figure 2: The outage probability against SNR for different number of the
secondary users. The users’ channel gains are assumed to be i.i.d. Rayleigh
fading. $R_{0}^{th}$ = 0.2 bit per channel uses (BPCU), $R_{s}^{th}$ = 1 BPCU,
$\Omega_{0}=\Omega_{m}=1$ and $K=1$.

In Fig.2, the impact of $M$ on the outage probability versus SNR is
investigated for the proposed system where $[M]:[1,5]$ and $[K]:[1]$. In case
$K$ is 1, as well as increasing $M$ degrades the outage performance, it is
seen that the outage performance of QoS-based SIC with and without algorithms
is better compared to CSI-based SIC. Moreover, the effect of the error floor
in CSI-based SIC is dramatically seen compared to QoS-based SIC with and
without algorithms. In case $M$ is 5, without the use of any proposed
algorithms, the outage probability of CSI-based SIC is obtained by selecting
the secondary user with maximum channel gain between the antenna and secondary
users, while the outage probability of QoS-based SIC is obtained by selecting
the secondary user with minimum channel gain between the antenna and secondary
users [18]. It can be seen that the analysis results and the exhaustive
analysis are in agreement with the simulation results. The simulation results
of the outage probability of the QoS-based SIC without the proposed algorithms
are verified through the closed-form expression of $\mathrm{P}_{A}^{QoS}$
obtained in the Appendix.

Figure 3: The outage probability against SNR for different number of antennas.
The users’ channel gains are assumed to be i.i.d. Rayleigh fading.
$R_{0}^{th}$ = 0.2 BPCU, $R_{s}^{th}$ = 1 BPCU, $\Omega_{0}=\Omega_{m}=1$, and
$M=6$.

Fig.3 depicts the impact of the number of antennas in the base station on the
outage probability versus SNR for the proposed system where $[M]:[6]$ and
$[K]:[2,4,7]$. In case $M$ is 6, it can be seen that increasing $K$
dramatically degrades the outage probability. When $K$ is 2, although it is
seen that the error floor begins to occur at high SNR, the effect of the error
floor decreases as the number of antennas increases. The outage performance of
QoS-based SIC with Algorithm (2) is better compared with Algorithm (1)
especially. It is seen that the simulation results are verified by the
theoretical analysis for Algorithm 1 and the exhaustive analysis for Algorithm
2.

In Fig.4, the impact of $M$ on the outage probability versus SNR is
investigated for the proposed system where $[M]:[1,4,6]$ and $[K]:[6]$. In
case $K$ is 6, as well as increasing $M$ degrades the outage performance, it
is seen that the effect of the error floor is significantly reduced. Also,
increasing $M$ dramatically degrades the outage probability for both
Algorithms. As in Fig. 3, the outage performance of QoS-based SIC with
Algorithm (2) is better than compared with Algorithm (1). Moreover, it can be
seen that the analysis results and the exhaustive analysis are in agreement
with the simulation results.

In Fig.5, the effect of $\sigma_{e}^{2}$ in case imperfect CSI in the outage
probability against SNR is investigated compared to perfect CSI for the
proposed system where $K$ and $M$ is 5. In the case of imperfect CSI, the
outage probability in the NOMA network with the QoS-based SIC increases for
Algorithm (2) compared to the perfect CSI case. Although there is a channel
estimation error in the case of imperfect CSI, it is observed that Algorithm
(2) works quite well.

Figure 4: The outage probability against SNR for different number of the
secondary users. The users’ channel gains are assumed to be i.i.d. Rayleigh
fading. $R_{0}^{th}$ = 0.2 BPCU, $R_{s}^{th}$ = 1 BPCU,
$\Omega_{0}=\Omega_{m}=1$, and $K=6$.

In Fig.6, in case of bad channel conditions between primary user and antennas
compared to secondary users, the outage probability against $\Omega_{0}$ for
the proposed system is investigated, where $[M]:[5]$ and $[K]:[3,5]$. As
$\Omega_{0}$ increases, the outage probability in the NOMA network with the
QoS-based SIC for both proposed algorithms increases compared to CSI-based
SIC. As in Fig. 4, the outage performance of QoS-based SIC with Algorithm (2)
is better than compared with Algorithm (1). Increasing the number of antennas
significantly improves the outage performance of QoS-based SIC compared to
CSI-based SIC. In case $K$ is 3, the outage performance of the QoS-based SIC
is approximately equal to the outage performance of CSI-based SIC for both
proposed algorithms when $\Omega_{0}$ is about 0.5. However, in case $K$ is 5,
the outage performance of the QoS-based SIC for both proposed algorithms is
equal to the outage performance of CSI-based SIC for both proposed algorithms
when $\Omega_{0}$ is less than 0.5. It can be seen that the analysis results
and the exhaustive analysis are in agreement with the simulation results.

##  V Conclusion

In this paper, we have proposed an uplink CR-NOMA system with QoS-based SIC
consisting of multiple users and the base station with a multi-antenna. The
outage performance of this system is aimed to improve by selecting the antenna
and the secondary user over two different proposed algorithms for QoS-based
SIC. Also, the effect of the error floor is aimed to decrease for this system
with QoS-based SIC by selecting the antenna and the secondary user in case of
multi-user multi-antenna. On the other hand, to observe the effect of
selecting SIC decoding order on the system performance, the outage performance
of QoS-based SIC through these algorithms has been compared with CSI-based SIC
over the proposed system. The exact expression of the outage probability of
this proposed system model with QoS-based SIC for Algorithms (1) has been
derived in closed form over Rayleigh fading channels. In addition, the outage
performance of this system with QoS-based SIC for Algorithms (2) has been
verified by the exhaustive analysis. In the case base station with a single
antenna, the closed-form expression of the outage probability of this system
with QoS-based SIC by not its use of proposed algorithms has also been derived
and compared with proposed algorithms. In the case of multi-antenna in the
base station, the proposed algorithms for QoS-based SIC have provided to
degrade the outage probability effectively. Also, the outage probability is
degraded by increasing the number of secondary users and antenna and the error
floor is significantly decreased. Increasing the number of antennas results in
a more effective increase in the outage performance difference between
Algorithms (1) and Algorithms (2) compared to increasing the number of
secondary users.

Figure 5: The outage probability against SNR in case of imperfect CSI. The
users’ channel gains are assumed to be i.i.d. Rayleigh fading. $R_{0}^{th}$ =
0.2 BPCU, $R_{s}^{th}$ = 1 BPCU, $\Omega_{0}=\Omega_{m}=1$, and $K=M=4$.

## Appendix

Utilizing the secondary user signal with the weakest channel gain among $M$
secondary users simply to minimize the performance degradation of the primary
user if the base station has only one antenna in the proposed network, the
expression of outage probability can be written as

$\leavevmode\resizebox{465.06001pt}{}{$\mathrm{P}_{A}^{QoS}=\underbrace{\mathrm{Pr}\left(\frac{X}{1+Y_{1}}<\gamma_{0}\right)}_{K_{1}}+\underbrace{\mathrm{Pr}\left(\frac{X}{1+Y_{1}}>\gamma_{0},Y_{1}<\gamma_{s}\right)}_{K_{2}}$},$
(A.1)

where $X=\bar{\gamma}|h_{0}|^{2}$ and $Y_{m}=\bar{\gamma}|h_{m}|^{2}$. Also,
the PDFs of the unordered variable $X$ and $Y$ are
$f_{X}(x)=\frac{1}{\tilde{\Omega}_{0}}e^{-\frac{x}{\tilde{\Omega}_{0}}}$,
$f_{Y_{m}}(y)=\frac{1}{\tilde{\Omega}_{m}}e^{-\frac{y}{\tilde{\Omega}_{m}}}$,
respectively. It is assumed that $|h_{1}|^{2}\leq|h_{m}|^{2}\leq|h_{M}|^{2}$
in this part. In what follows, $K_{1}$ and $K_{2}$ are addressed to write the
closed-form expression of $\mathrm{P}_{A}^{QoS}$ given in (A.1). Firstly, the
expression of $K_{1}$ can be given as

$K_{1}=\int_{0}^{\infty}F_{X}\left(\gamma_{0}+\gamma_{0}y\right)f_{Y_{1}}(y)\mathrm{d}y.$
(A.2)

To calculate the expression of $K_{1}$,
$F_{X}\left(\gamma_{0}+\gamma_{0}y\right)$ and $f_{Y_{1}}(y)$ need to be
evaluated first. Firstly, $F_{X}\left(\gamma_{0}+\gamma_{0}y\right)$ is equal
to $1-e^{-\frac{\left(y+1\right)\gamma_{0}}{\tilde{\Omega}_{0}}}$. With the
aid of [27], the PDF of the ordered variable $Y_{1}$ can be calculated from
the CDF of the ordered variable $Y_{1}$, $F_{Y_{1}}(y)$ is given by

$F_{Y_{1}}(y)=M\sum_{s=0}^{M-1}\frac{\left(-1\right)^{s}}{1+s}\binom{M-1}{s}\left(1-e^{-\frac{y}{\tilde{\Omega}_{1}}}\right)^{s+1},$
(A.3)

where $\tilde{\Omega}_{1}=\bar{\gamma}\Omega_{1}$. Using (A.3), the PDF of the
ordered variable $Y_{1}$ can be written as

$\displaystyle f_{Y_{1}}(y)$
$\displaystyle=\frac{M}{\tilde{\Omega}_{1}}\sum_{s=0}^{M-1}\left(-1\right)^{s}\binom{M-1}{s}e^{-\frac{y}{\tilde{\Omega}_{1}}}\left(1-e^{-\frac{y}{\tilde{\Omega}_{1}}}\right)^{s}.$
(A.4)

Hence, $K_{1}$ can be rewritten as

$\displaystyle K_{1}$
$\displaystyle=1-e^{-\frac{\gamma_{0}}{\tilde{\Omega}_{1}}}\int_{0}^{\infty}e^{-\frac{-y\gamma_{0}}{\tilde{\Omega}_{1}}}f_{Y_{1}}(y)\mathrm{d}y$
$\displaystyle=1-\frac{M}{\tilde{\Omega}_{1}}e^{-\frac{\gamma_{0}}{\tilde{\Omega}_{1}}}\sum_{s=0}^{M-1}\sum_{p=0}^{s}\frac{\left(-1\right)^{s+p}\binom{M-1}{s}\binom{s}{p}}{\frac{(p+1)}{\tilde{\Omega}_{1}}+\frac{{\gamma}_{0}}{\tilde{\Omega}_{0}}}.$
(A.5)

Secondly, using (A.3) and (Appendix), the closed-form expression of outage
probability of $K_{2}$ can be calculated as

$\displaystyle K_{2}=$
$\displaystyle\left(1-K_{1}\right)\left[\int_{0}^{\gamma_{s}}f_{Y_{1}}\left(y\right)\mathrm{d}y\right]$
(A.6) $\displaystyle=$
$\displaystyle\left(1-K_{1}\right)\left[M\sum_{s=0}^{M-1}\frac{\left(-1\right)^{s}}{1+s}\binom{M-1}{s}\left(1-e^{-\frac{\gamma_{s}}{\tilde{\Omega}_{1}}}\right)^{s+1}\right].$

Finally, the closed form expression of $\mathrm{P}_{A}^{QoS}$ is obtained by
writing $K_{1}$ and $K_{2}$ in (A.1).

Figure 6: The outage probability against $\Omega_{0}$ for different number of
antennas. The users’ channel gains are assumed to be i.i.d. Rayleigh fading.
$R_{0}^{th}$ = 0.2 BPCU, $R_{s}^{th}$ = 1 BPCU, $\Omega_{m}=1$, $\bar{\gamma}$
= 10dB, and $M=5$.

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