MacWilliams Type Identities for Linear Block Codes on Certain Pomsets

The study of codes endowed with a metric other than the Hamming metric gained momentum since 1990’s with the introduction of poset metric by Brialdi et al. (see [References]). Feng et al. in [References] introduced block metric and studied MDS block codes. Poset block metric was introduced by Alves et al. in [References] unifying poset metric and block metric.

More recently in [References], Selvaraj and Sudha generalized the poset metric structure to a pomset metric structure. They introduced pomset metric and initialized the study of codes equipped with pomset metric. The concept of order ideals of a pomset is proposed and pomset metric is defined. We also refer the reader to [References, References] for two general metrics in poset space.

Weight distribution of a code describes the number of codewords of each possible weight. The MacWilliams identity for linear codes over finite fields is one of the most important identities in coding theory (see [References]). It expresses a connection between the weight enumerator of a linear codes and its dual code.

Kim and Oh classified all poset structures that admit the MacWilliams identity, and derived the MacWilliams identities for poset weight enumerators corresponding to such posets (see [References]). It is proved that being hierarchical is a necessary and sufficient condition for a poset to admit the MacWilliams identity. They also derived an explicit relation between the $P$-weight distribution of a hierarchical poset code and the $\bar{P}$-weight distribution of the dual code. Extending their observations, Pinheiro and Firer proved that a poset-block space admits a MacWilliams-type identity if and only if the poset is hierarchical, and at any level of the poset, all the blocks have the same dimension (see [References]). They explicitly stated the relation between the weight enumerators of a code and its dual when the poset-block admits the MacWilliams-type identity.

For codes over $\mathbb{Z}_{m}$, pomset metric is a generalization of Lee metric when pomset is taken to be an antichain; in some sense, it is a generalization to poset metric as well. Sudha and Selvaraj defined pomset weight enumerator of a code $\mathcal{C}$ and established MacWilliams type identities for linear codes with respect to certain pomsets (see [References]). The identities for a particular type of linear codes are established by considering direct and ordinal sum of pomsets on them. For pomset block codes, it is natural to attempt to obtain some kind of MacWilliams identities for block codes in certain pomset metric. In this paper, we generalize the MacWilliams type identity given in [References] for pomset spaces to pomset-block spaces.

The paper is organized as follows, Section 2 contains basic notions of pomset block metric over $\mathbb{Z}_{m}^{n}$ and defines weight enumerator for a linear pomset block code. In Section 3, we consider the relationship between the weight distribution of a pomset block code and its dual when the pomset is a chain pomset. In particular, an explicit relation is derived between the $\mathbb{P}$-weight distribution of a pomset block code and the $\widetilde{\mathbb{P}}$-weight distribution of the dual code when $\mathbb{Z}_{m}$ is a field and all blocks have dimension 2. We also give same examples to illustrate our conclusion. In Section 4, we give MacWilliams type identities on direct and ordinal sums of general pomsets. Lastly, we summarize our results and raise a question for further research.

In this section, we introduce some basic notations and useful results of a pomset block metric.

A collection of elements which may contain duplicates is called a multiset (in short, mset). Girish and John defined a multiset relation and explored some of basic properties (see [References] and [References]).

Let $X$ be a set formally. A mset $M$ drawn from the set $X$ is represented by a function count $C_{M}:X\rightarrow\mathbb{N}$ where $\mathbb{N}$ represents the set of non-negative integers. For each $a\in X$, $C_{M}(a)$ indicates the number of occurrences of the element $a$ in $M$.

An element $a\in X$ appearing $p$ times in $M$ is denoted by $p/a\in M$ and thus $C_{M}(a)=p$. If we consider $k/a\in M$, the value of $k$ satisfies $k\leq p$. The mset drawn from the set $X=\{a_{1},a_{2},\ldots,a_{n}\}$ is represented as $M=\{p_{1}/a_{1},p_{2}/a_{2},\ldots,p_{n}/a_{n}\}$. The cardinality of an mset $M$ drawn from $X$ is $|M|=\sum_{a\in X}C_{M}(a)$. The root set of $M$ denoted by $M^{*}$ is defined as $M^{*}=\{a\in X:C_{M}(a)>0\}$.

Let $M_{1}$ and $M_{2}$ be two msets drawn from a set $X$. We call $M_{1}$ a submset of $M_{2}$ ($M_{1}\subseteq M_{2}$) if $C_{M_{1}}(a)\leq C_{M_{2}}(a)$ for all $a\in X$. The union of $M_{1}$ and $M_{2}$ is an mset denoted by $M=M_{1}\cup M_{2}$ such that for all $a\in X$, $C_{M}(a)=\text{max}\{C_{M_{1}}(a),C_{M_{2}}(a)\}$.

Let $M_{1}$ and $M_{2}$ be two msets drawn from $X$, the Cartesian product of $M_{1}$ and $M_{2}$ is also an mset defined as

Denote by $C_{1}(a,b)$ the count of the first coordinate in the ordered pair $(a,b)$ and by $C_{2}(a,b)$ the count of the second coordinate in the ordered pair $(a,b)$.

A submset $R$ of $M\times M$ is said to be an mset relation on $M$ if every member $(p/a,q/b)$ of $R$ has count $C_{1}(a,b)\cdot C_{2}(a,b)$. An mset relation $R$ on an mset $M$ is said to be reflexive if $m/a\ R\ m/a$ for all $m/a\in M$; antisymmetric if $m/a\ R\ n/b$ and $n/b\ R\ m/a$ imply $m=n$ and $a=b$; transitive if $m/a\ R\ n/b$ and $n/b\ R\ k/c$ imply $m/a\ R\ k/c$. An mset relation $R$ is called a partially ordered mset relation (or order relation) if it is reflexive, antisymmetric and transitive. The pair $(M,R)$ is known as a partially ordered multiset (pomset) denoted by $\mathbb{P}$.

Let $\mathbb{P}=(M,R)$ and $m/a\in M$. Then $m/a$ is a maximal element of $\mathbb{P}$ if there exists no $n/b\in M\ (b\neq a)$ such that $m/a\ R\ n/b$; $m/a$ is a minimal element if there exists no $n/b\in M\ (b\neq a)$ such that $n/b\ R\ m/a$. $\mathbb{P}$ is called a chain if every distinct pair of points from $M$ is comparable in $\mathbb{P}$. $\mathbb{P}$ is called an anti-chain if every distinct pair of points from $M$ is incomparable in $\mathbb{P}$.

A submset $I$ of $M$ is called an order ideal (or simply an ideal) of $\mathbb{P}$ if $k/a\in I$ and $q/b\ R\ k/a$ $(b\neq a)$ imply $q/b\in I$. An ideal generated by an element $k/a\in M$ is defined as

An ideal generated by a submset $S$ of $M$ is defined by $\langle S\rangle=\bigcup\limits_{k/a\in S}\langle k/a\rangle$.

For a given pomset $\mathbb{P}=(M,R)$, the dual pomset $\widetilde{\mathbb{P}}=(M,\widetilde{R})$ of $\mathbb{P}$ is given by:

$\mathbb{P}$ and $\widetilde{\mathbb{P}}$ have the same underlying set $M$ and $p/a\ R\ q/b$ in $\mathbb{P}$ if and only if $q/b\ \widetilde{R}\ p/a$ in $\widetilde{\mathbb{P}}$.

Note that $\mathbb{P}$ is a chain pomset implies that $\widetilde{\mathbb{P}}$ is a chain pomset.

Consider $\mathbb{Z}_{m}=\{0,1,\ldots,m-1\}$, the ring of integers modulo $m$. We consider a pomset $\mathbb{P}$ defined on an mset $M=\left\{\left\lfloor\frac{m}{2}\right\rfloor/1,\left\lfloor\frac{m}{2}\right\rfloor/2,\ldots,\left\lfloor\frac{m}{2}\right\rfloor/s\right\}$.

Let $\pi:[s]\rightarrow\mathbb{N}$ be a map such that $n=\sum\limits_{i=1}\limits^{s}\pi(i)$. The map $\pi$ is said to be a labeling of the pomset $\mathbb{P}$, and the pair $(\mathbb{P},\pi)$ is called a pomset block structure over $[s]$. Denote $\pi(i)$ by $k_{i}$ and take $V_{i}$ as free $\mathbb{Z}_{m}$-module $\mathbb{Z}_{m}^{k_{i}}$ for all $1\leq i\leq s$. Define $V$ as

which is isomorphic to $\mathbb{Z}_{m}^{n}$. Each $\boldsymbol{u}\in V$ can be written as

where $\boldsymbol{u_{i}}=(u_{i1},u_{i2},\ldots,u_{ik_{i}})\in\mathbb{Z}_{m}^{k_{i}}$, $1\leq i\leq s$. For $a\in\mathbb{Z}_{m}$, Lee weight $w_{L}(a)$ of $a$ is minimum of $a$ and $m-a$. The Lee block support of $\boldsymbol{u}\in V$ is defined as

where

For a linear $(\mathbb{P},\pi)$-code $\mathcal{C}$, the $(\mathbb{P},\pi)$-weight enumerator for $\mathcal{C}$ is the polynomial

where $A_{i,(\mathbb{P},\pi)}(\mathcal{C})=\left|\{\boldsymbol{u}\in\mathcal{C}:w_{(\mathbb{P},\pi)}(\boldsymbol{u})=i\}\right|$.

Let $[a]=\{1,2,\ldots,a\}$ and $[a,b]$ be the set of all integers between $a$ and $b$. In this section, we will derive the MacWilliams type identity for linear block codes in the chain pomset metric. Without loss of generality, we define the pomset $\mathbb{P}=(M,R)$ on the multiset $M=\{\lfloor\frac{m}{2}\rfloor/1,\ldots,\lfloor\frac{m}{2}\rfloor/s\}$ whose mset relation is given by

Let $(\mathbb{P},\pi)$ be a pomset block structure on $V$ where $\pi$ is a labeling of $\mathbb{P}$ such that $\sum\limits_{i=1}^{s}\pi(i)=n$. Suppose that $\boldsymbol{u}=(\boldsymbol{u_{1}},\ldots,\boldsymbol{u_{s}})\in V$ such that $\boldsymbol{u_{i}}\neq\textbf{0}\in\mathbb{Z}_{m}^{\pi(i)}$ and $(\boldsymbol{u_{i+1}},\boldsymbol{u_{i+2}},\ldots,\boldsymbol{u_{s}})=\textbf{0}\in\mathbb{Z}_{m}^{\pi(i+1)+\cdots+\pi(s)}$. By the definition of $\mathbb{P}$, we have that $w_{(\mathbb{P},\pi)}(\boldsymbol{u})=\left\lfloor\frac{m}{2}\right\rfloor(i-1)+w_{(L,\pi)}(\boldsymbol{u_{i}})$. Given a linear $(\mathbb{P},\pi)$-code $\mathcal{C}$, we define

and

Let $\mathcal{R}$ be a finite ring. Recall that an additive character $\chi$ on $\mathcal{R}$ is just a group homomorphism from the additive group $\mathcal{R}$ into the multiplicative group $\mathbb{C}^{*}$. The set of all additive character of $\mathcal{R}$ forms a group $\hat{\mathcal{R}}$, called the character group whose group operation is the pointwise multiplication of characters. Moreover, $\hat{\mathcal{R}}$ is a right $\mathcal{R}$-module with the function $\mathcal{R}\times\hat{\mathcal{R}}\rightarrow\hat{\mathcal{R}}$ given by $(a,\chi)=\chi_{a}$ where $\chi_{a}\in\hat{\mathcal{R}}$ such that $\chi_{a}(b)=\chi(ab)$ for all $\chi\in\hat{\mathcal{R}}$ and $a,b\in\mathcal{R}$. A character $\chi$ of $\mathcal{R}$ is a right generating character if the mapping $\phi:\mathcal{R}\rightarrow\hat{\mathcal{R}}$ given by $\phi(r)=\chi_{r}$ is an isomorphism of $\mathcal{R}$-modules. See [References] for detailed discussion on additive characters.

The following lemmas are easy consequences of Lemma 3.1.

Let $f$ be a complex-valued function defined on $\mathcal{R}^{n}$ and $\chi$ be a generating character of $\mathcal{R}$. The Fourier transform of $f$ is