# The MacWilliams Theorem for Four-Dimensional Modulo Metrics

Mehmet Özen, Murat Güzeltepe
Department of Mathematics, Sakarya University, TR54187 Sakarya, Turkey

###### Abstract

In this paper, the MacWilliams theorem is stated for codes over finite field
with four-dimensional modulo metrics.

AMS Classification: 94B05, 94B60

Keywords: MacWilliams theorem, Block codes, Weight enumerator, Quaternion
Mannheim metric.

##  1 Introduction

The MacWilliams theorem is one of the most important theorems in coding
theory. It is well known that two of the most famous results in block code
theory are MacWilliams Identity Theorem end Equivalence Theorem [1, 2]. Given
the weight enumerator of an code, the MacWilliams theorem ensure one to obtain
the weight enumerator of the dual code . The MacWilliams theorem very useful
since weight distribution of high rate codes can be obtained from low rate
codes. A well known version of the MacWilliams theorem for codes with respect
to Hamming weight was presented in [3]. The more general version of this
theorem are less often used in practical applications. The impact of this
theorem for practical as well as theoretical purposes is well known, see for
instance [3, Chs. 11.3, 6.5, and 19.2]. In [4], the MacWilliam theorem proved
for codes over finite fields with two-dimensional modulo metric.

In this study, we utilize the MacWilliam theorem for complete weight
enumerators to obtain the MacWilliams theorem for codes over quaternion
integers (QI). The Hamilton quaternion algebra is defined as follows.

###### Definition 1

Let $\mathcal{R}$ be the field of real numbers. The Hamilton Quaternion
Algebra over $\mathcal{R}$ denoted by $H[\mathcal{R}]$ is the associative
unital algebra given by the following representation:

i)$H[\mathcal{R}]$ is the free $\mathcal{R}$ module over the symbols
$1,i,j,k$, that is, $H[\mathcal{R}]=\\{a_{0}+a_{1}i+$
$a_{2}j+a_{3}k:\;a_{0},a_{1},a_{2},a_{3}\in R\\}$;

ii)1 is the multiplicative unit;

iii) $i^{2}=j^{2}=k^{2}=-1$;

iv) $ij=-ji=k,\;ik=-ki=j,\;jk=-kj=i$ [5].

If $q=a_{0}+a_{1}i+a_{2}j+a_{3}k$ is a quaternion integer, its conjugate
quaternion is $\overline{q}=a_{0}-(a_{1}i+a_{2}j+a_{3}k)$. The norm of $q$ is
$N(q)=q.\overline{q}=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}$, which is
multiplicative, that is, $N(q_{1}q_{2})=N(q_{1})N(q_{2})$. It should be noted
that quaternions are not commutative. The ring of the integers of the
quaternions is
$\;H[\mathcal{Z}]=\left\\{{a_{0}+a_{1}i+a_{2}j+a_{3}k:\;a_{0},a_{1},a_{2},a_{3}\in\mathcal{Z}}\right\\}$.
Let $H[\mathcal{Z}]_{\pi}$ be residue class of $H[\mathcal{Z}]$ modulo $\pi$,
where $\pi$ is prime quaternion integer. The set obtained form the elements of
$H[\mathcal{Z}]_{\pi}$ obtains the elements which by the remainders from right
dividing (or left dividing) the elements of $H[\mathcal{Z}]$ by the element
$\pi$. For example, let $p=3,\pi=1+i+j$ then we get
$H[\mathcal{Z}]_{\pi}=\left\\{{\mp 1,\mp i,\mp j,\mp k}\right\\}$. Also
$H[\mathcal{Z}]_{\pi}$ has $N(\pi)^{2}$ elements [6]. More information which
is related with the arithmetic properties of $H[\mathcal{Z}]$ can be found in
[5, pp. 57-71]. The quaternion Mannheim metric also called Lipschitz metric
was defined in [6, 7]. Let
$\alpha-\beta\equiv\delta=a_{0}+a_{2}i+a_{2}j+a_{3}k\,(\bmod\;\pi)$. Then the
weight of $\delta$ which is denoted by $w_{QM}(\delta)$ is equal
$\left|{a_{0}}\right|+\left|{a_{2}}\right|+\left|{a_{2}}\right|+\left|{a_{3}}\right|$.
The distance between $\alpha$ and $\beta$ was defined as
$d_{QM}(\alpha,\beta)=w_{QM}(\delta)$.
Now we recall some notation and definitions on characters and weight
enumerators needed in this paper. Let $\gamma$ be an element of the Galois
field $GF(p^{m})$. Using the primitive element $\alpha$, $\gamma$ can be
represented as $\gamma=\sum\nolimits_{t=0}^{m-1}{g_{t}\alpha^{t}}$ with
$g_{t}$ from $GF(p)$. The character $\chi_{1}(\gamma)$ is defined using the
primitive complex $p-th$ root $\xi$:

$\chi_{1}(\gamma)=\xi^{g_{0}}$

where
$\xi=\exp({{2\pi\sqrt{-1}}\mathord{\left/{\vphantom{{2\pi\sqrt{-1}}p}}\right.\kern-1.2pt}p})$,
$\pi=3,14...$
The complete weight enumerator classifies the codewords of a linear code
according to the number of times each field element $\omega_{t}$ appears in
the codeword. The composition of a vector
$u=\left({\begin{array}[]{*{20}c}{u_{0},}&{u_{1},}&\cdots&{,u_{n-1}}\\\
\end{array}}\right)$ denoted by $comp(u)$ is given by
$s=\left({\begin{array}[]{*{20}c}{s_{0},}&{s_{1},}&\cdots&{,s_{q-1}}\\\
\end{array}}\right)$, where $s_{t}$ is the number of components $u_{t}$ equal
to $\omega_{t}$. Note that there exist a group homomorphism between
$GF(p^{2})$ and $H[\mathcal{Z}]_{\pi}$ using a rational mapping. For example,
assume that $p=3$ then $\pi=1+i+j$,
$GF(p^{2})=\left\\{{0,1,\alpha,\alpha^{2},...,\alpha^{7}}\right\\}$ and
$H[\mathcal{Z}]_{\pi}=\left\\{{0,1,-1,i,-i,j,-j,k,-k}\right\\}$ where
$\alpha^{2}=\alpha+1,\;\alpha^{8}=1$. We obtain a group homomorphism mapping 0
to 0, 1 to 1, $\alpha$ to $i$, $2\alpha$ to $-i$, $2+2\alpha$ to $j$,
$1+\alpha$ to $-j$, $2\alpha+1$ to $k$, $\alpha+2$ to $-k$.

###### Definition 2

The composition of
$u=\left({\begin{array}[]{*{20}c}{u_{0},}&{u_{1},}&\cdots&{,u_{n-1}}\\\
\end{array}}\right)$, denoted by $comp(u)$, is
$s=\left({\begin{array}[]{*{20}c}{s_{0},}&{s_{1},}&\cdots&{s_{q-1}}\\\
\end{array}}\right)$ where $s_{t}=s_{t}(u)$ is the number of components
$u_{t}$ equal to $\omega_{t}$. Thus it is obtain

$\sum\limits_{t=0}^{q-1}{s_{t}(u)=n}.$

Let $C$ be a linear $[n,k]$ code over $GF(p)$. Then the complete weight
enumerator of $C$

$W_{C}(z_{0},z_{1},\cdots,z_{q-1})=\sum\limits_{c\in
C}{\left({\prod\limits_{t=0}^{q-1}{z_{t}^{s_{t}(u)}}}\right)}$

where $z_{t}$ are indeterminates and the sum extends over all compositions.

The MacWilliams theorem for complete weight enumerators [3, pp.143-144, Thm
10] then states:

###### Theorem 1

The complete weight enumerator of the dual code $C^{\bot}$ can be obtained
from the complete weight enumerator of the code $C$ by replacing each $z_{t}$
by

$\sum\limits_{s=0}^{q-1}{\chi_{1}(\omega_{t}\omega_{s})}z_{s}$

and dividing the result by the cardinality of $C$ which is denoted by
$\left|C\right|$.

##  2 The MacWilliams Theorem for codes over Quaternion Integers

Let $GF(q)$ be a finite field with $q=p^{m}$. The field $GF(q)$ is partitioned
as follows:

$GF(q)=\left\\{0\right\\}\cup G_{1}\cup G_{2}\cup G_{3}\cup G_{4}\cup
G_{5}\cup G_{6}\cup G_{7}\cup G_{8}.$

We set $\omega_{0}=0$. $G_{1}$ contains
${{\left({q-1}\right)}\mathord{\left/{\vphantom{{\left({q-1}\right)}8}}\right.\kern-1.2pt}8}$
elements
$\omega_{t},\,t=1,2,...,{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}$
in a fixed way such that for
$t=1,2,...,{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}$
we have

$\begin{array}[]{l}G_{2}=\omega_{2}G_{1},\;\omega_{2}\notin G_{1},\\\
G_{3}=\omega_{3}G_{1},\;\omega_{3}\notin G_{1}\cup G_{2},\\\
G_{4}=\omega_{4}G_{1},\;\omega_{4}\notin G_{1}\cup G_{2}\cup G_{3},\\\
G_{5}=\omega_{5}G_{1},\;\omega_{5}\notin G_{1}\cup G_{2}\cup G_{3}\cup
G_{4},\\\ G_{6}=\omega_{6}G_{1},\;\omega_{6}\notin G_{1}\cup G_{2}\cup
G_{3}\cup G_{4}\cup G_{5},\\\ G_{7}=\omega_{7}G_{1},\;\omega_{7}\notin
G_{1}\cup G_{2}\cup G_{3}\cup G_{4}\cup G_{5}\cup G_{6},\\\
G_{8}=\omega_{8}G_{1},\;\omega_{8}\notin G_{1}\cup G_{2}\cup G_{3}\cup
G_{4}\cup G_{5}\cup G_{6}\cup G_{7}.\\\ \end{array}.$

The quaternion Mannheim weight of a vector $u$ over $GF(p)$ is defined as
$quaternionic(u)=\left({\begin{array}[]{*{20}c}{g_{0},}&{g_{1},}&\cdots&{,g_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}}\\\
\end{array}}\right).$ Note that the quaternion integer enumerator does not
distinguish between the eight elements $\mp\omega,\mp i\omega,\mp j\omega,\mp
k\omega.$ The complete weight enumerator of the dual code $C^{\bot}$ from the
complete weight enumerator of the code $C$ over $H[\mathcal{Z}]_{\pi}$
obtained as follows:

###### Theorem 2

The quaternion integer (QI) weight enumerator of the dual code $C^{\bot}$ can
be obtained from QI weight enumerator of $C$ by replacing $z_{1}$ by

$z_{0}+\sum\limits_{s=1}^{{{\left({q-1}\right)}\mathord{\left/{\vphantom{{\left({q-1}\right)}8}}\right.\kern-1.2pt}8}}{\left[\begin{array}[]{l}\chi_{1}(\omega_{1}\omega_{s})+\chi_{1}(-\omega_{1}\omega_{s})+\chi_{1}(i\omega_{1}\omega_{s})+\chi_{1}(-i\omega_{1}\omega_{s})+\chi_{1}(j\omega_{1}\omega_{s})\\\
+\chi_{1}(-j\omega_{1}\omega_{s})+\chi_{1}(k\omega_{1}\omega_{s})+\chi_{1}(-k\omega_{1}\omega_{s})\\\
\end{array}\right]}z_{s}=z_{0}+$
$[\chi_{1}(\omega_{1}\omega_{1})+\chi_{1}(-\omega_{1}\omega_{1})+\chi_{1}(i\omega_{1}\omega_{1})+\chi_{1}(-i\omega_{1}\omega_{1})+\chi_{1}(j\omega_{1}\omega_{1})+\chi_{1}(-j\omega_{1}\omega_{1})+\chi_{1}(k\omega_{1}\omega_{1})+\chi_{1}(-k\omega_{1}\omega_{1})]z_{1}+\cdots$
$+\left[\begin{array}[]{l}\chi_{1}(\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
+\chi_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}(-j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
\end{array}\right]z_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}},$

$z_{2}$ by

$[\chi_{1}(\omega_{1}\omega_{1})+\chi_{1}(-\omega_{1}\omega_{1})+\chi_{1}(i\omega_{1}\omega_{1})+\chi_{1}(-i\omega_{1}\omega_{1})+\chi_{1}(j\omega_{1}\omega_{1})+\chi_{1}(-j\omega_{1}\omega_{1})+\chi_{1}(k\omega_{1}\omega_{1})+\chi_{1}(-k\omega_{1}\omega_{1})]z_{2}+\cdots$
$+\left[\begin{array}[]{l}\chi_{1}(\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
+\chi_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}(-j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
\end{array}\right]z_{1}...$

and using the same argument, shifting the coefficients of
$z_{1},z_{2},\cdots,z_{{{\left({q-1}\right)}\mathord{\left/{\vphantom{{\left({q-1}\right)}8}}\right.\kern-1.2pt}8}}$,
$z_{{{\left({q-1}\right)}\mathord{\left/{\vphantom{{\left({q-1}\right)}8}}\right.\kern-1.2pt}8}}$
by

$[\chi_{1}(\omega_{1}\omega_{1})+\chi_{1}(-\omega_{1}\omega_{1})+\chi_{1}(i\omega_{1}\omega_{1})+\chi_{1}(-i\omega_{1}\omega_{1})+\chi_{1}(j\omega_{1}\omega_{1})+\chi_{1}(-j\omega_{1}\omega_{1})+\chi_{1}(k\omega_{1}\omega_{1})+\chi_{1}(-k\omega_{1}\omega_{1})]z_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}+\cdots$
$+\left[\begin{array}[]{l}\chi_{1}(\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-i\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
+\chi_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}(-j\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})+\chi_{1}(-k\omega_{1}\omega_{{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}})\\\
\end{array}\right]z_{\left({{{(q-1)}\mathord{\left/{\vphantom{{(q-1)}8}}\right.\kern-1.2pt}8}}\right)-1}.$

The proof is immediately obtained from MacWilliams theorem for complete weight
enumerators above.

###### Example 1

Let $p=3,\;\pi=1+i+j+k$. Then
$H[\mathcal{Z}]_{\pi}=\left\\{{0,1,-1,i,-i,j,-j,k,-k}\right\\}$. Let us
consider $[2,1,2]$ \- code $C$ over $GF(9)=H[\mathcal{Z}]_{\pi}$. Thus we get
$GF(9)=H[\mathcal{Z}]_{\pi}=G_{0}\cup G_{1}=\left\\{0\right\\}\cup\left\\{{\mp
1,\mp i,\mp j,\mp k}\right\\},\omega_{0}=0,\;\omega_{1}=1$ Assume that the
code $C$ which is an left ideal of $H[\mathcal{Z}]_{\pi}\times
H[\mathcal{Z}]_{\pi}$ is generate by the matrix $(1,1)$. Then the complete
weight enumerator of $C$ is $w_{QM}(C)=z_{0}^{2}+8z_{1}^{2}$. Applying the QI
MacWilliams theorem means that to replace $z_{1}\to
z_{0}+\left({\xi^{1}+\xi^{2}+\xi^{0}+\xi^{0}+\xi^{2}+\xi^{1}+\xi^{1}+\xi^{2}}\right)z_{1}=z_{0}-z_{1}$.
$1+\xi^{1}+\xi^{2}=0$ since there is a group homomorphism between $GF(p^{2})$
and $H[\mathcal{Z}]_{\pi}$, where $\xi=e^{{{2\pi
i}\mathord{\left/{\vphantom{{2\pi i}3}}\right.\kern-1.2pt}3}},\;\pi=3,14...$
Thus the complete weight enumerator of the dual code $C^{\bot}$ is equal
$z_{0}^{2}+8z_{1}^{2}=w_{QM}(C)$.

###### Example 2

Let $p=5,\;\pi=2+i$. Then

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
H[\mathcal{Z}]_{\pi}=\left\\{0\right\\}\cup\left\\{{1,-1,i,-i,j,-j,k,-k}\right\\}\cup(1+j)\left\\{{1,-1,i,-i,j,-j,k,-k}\right\\}\cr
0.0pt{\hfil$\displaystyle\cup(1+k)\left\\{{1,-1,i,-i,j,-j,k,-k}\right\\}.\cr}}}$

Let us consider $[3,1,3]$-code over $GF(25)=H[\mathcal{Z}]_{2+i}$. Thus we
get, $\omega_{0}=0,\;\omega_{1}=1,\;\omega_{2}=1+\alpha\leftrightarrow
1+j,\;\omega_{3}=1+2\alpha\leftrightarrow 1+k$. Assume that the code $C$ which
is an left ideal of $H[\mathcal{Z}]_{\pi}\times H[\mathcal{Z}]_{\pi}$ is
generate by the matrix $\left({\begin{array}[]{*{20}c}1&1&1\\\
\end{array}}\right)$. Then the complete weight enumerator of $C$ is
$w_{QM}(C)=z_{0}^{3}+8z_{1}^{3}+8z_{2}^{3}+8z_{3}^{3}$. Applying the QI
MacWilliams theorem means that to replace

$z_{0}\to z_{0}+8z_{1}+8z_{2}+8z_{3},$ $\halign{\hbox
to\displaywidth{$\hfil\displaystyle#\hfil$}\cr 0.0pt{\hfil$\displaystyle
z_{1}\to
z_{0}+(\xi^{1}+\xi^{4}+\xi^{3}+\xi^{2}+\xi^{0}+\xi^{0}+\xi^{0}+\xi^{0})z_{1}+\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{2}\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{3},\cr}}}}$
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle z_{2}\to
z_{0}+(\xi^{1}+\xi^{4}+\xi^{3}+\xi^{2}+\xi^{0}+\xi^{0}+\xi^{0}+\xi^{0})z_{2}+\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{3}\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{1},\cr}}}}$
$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle z_{3}\to
z_{0}+(\xi^{1}+\xi^{4}+\xi^{3}+\xi^{2}+\xi^{0}+\xi^{0}+\xi^{0}+\xi^{0})z_{3}+\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{1}\cr
0.0pt{\hfil$\displaystyle+(\xi^{1}+\xi^{4}+\xi^{4}+\xi^{1}+\xi^{3}+\xi^{2}+\xi^{2}+\xi^{3})z_{2}.\cr}}}}$

$1+\xi^{1}+\xi^{2}+\xi^{3}+\xi^{4}=0$ since there is a group homomorphism
between $GF(5^{2})$ and $H[\mathcal{Z}]_{2+i}$, where $\xi=e^{{{2\pi
i}\mathord{\left/{\vphantom{{2\pi i}5}}\right.\kern-1.2pt}5}},\;\pi=3,14...$
Thus the complete weight enumerator of the dual code $C^{\bot}$ is equal

$\halign{\hbox to\displaywidth{$\hfil\displaystyle#\hfil$}\cr
0.0pt{\hfil$\displaystyle
z_{0}^{3}+24z_{0}z_{1}^{2}+24z_{0}z_{2}^{2}+24z_{0}z_{3}^{2}+24z_{1}^{3}+24z_{2}^{3}+24z_{3}^{3}\cr
0.0pt{\hfil$\displaystyle+48z_{1}^{2}z_{2}+48z_{1}z_{2}^{2}+48z_{2}z_{3}^{2}+48z_{1}^{2}z_{3}+48z_{1}z_{3}^{2}+48z_{2}^{2}z_{3}\cr
0.0pt{\hfil$\displaystyle+192z_{1}z_{2}z_{3}.\cr}}}}$

##  3 Conclusion

In this paper, we proved the MacWilliams for four-dimensional modulo metrics.
In fact, the quaternion Mannheim metric can be seen as a four-dimensional
generalization of the Lee metric. Also the quaternion Mannheim metric can be
seen as a four-dimensional generalization of the Mannheim metric. In other
words, if four-dimensional space is restricted to two-dimensional space then
results in [4] are obtained.

## References

  * [1] F. J. MacWilliams, ”Combinatorial Problems of Elementary Abelian Groups,” Ph.D. dissertation, Harvard Univ., Cambridge, MA, 1962.
  * [2] F. J. MacWilliams, ”A theorem on the distribution of weights in a systematic code,” Bell Syst. Tech. J., vol. 42, pp. 79-94, 1963.
  * [3] F. J. Macwilliams and N. J. Sloane, ”The Theory of Error Correcting Codes”, North Holland Pub. Co., 1977.
  * [4] K. Huber, ”The MacWilliams Theorem for Two-Dimensional Modulo Metrics,” AAECC, 41-48, 1997. (submitted, 2009).
  * [5] G. Davidoff, P. Sarnak, A. Valette, ”Elementary Number Theory, Group Theory, Ramanujan Graphs”, Cambridge University Pres, 2003.
  * [6] C. Martinez et al. ”Perfect Codes from Cayley Graphs over Lipschitz Integers,” IEEE Trans. Inform.Theory, vol. 55, pp. 3552-3562, August, 2009.
  * [7] M. zen and M. G zeltepe, ”Codes over Quaternion Integers”, e-print arXiv:0905.4160v1.