Relative Duality in MacWilliams Identity

Abstract

This work considers relations, which connect weight enumerator of a linear code $\mathcal{L} \subset \mathbb{F}_p^n$ and weight enumerator of code $\mathcal{K}$ , lying between $\mathcal{L}$ and $\mathbb{F}_p^n :\mathcal{L} \subseteq \mathcal{K} \subseteq \mathbb{F}_p^n$ . The connection is establish by code $\mathcal{L}^ \bot \left( \mathcal{K} \right)$ , which is dual code of code $\mathcal{L}$ in the code $\mathcal{K}$ . It has been show, that there is a code $\mathcal{L}^ \bot \left( \mathcal{K} \right)$ , which dimension is k−l, where l and k are dimensions of $\mathcal{L}$ and $\mathcal{K}$ respectively. These relations for weight enumerators generalise the MacWilliams relation, in which $\mathcal{K} = \mathbb{F}_p^n$.

The new relations are written for linear code of length 2n of code $\mathcal{L} = \left\{ {\left( {a\left| {a + b} \right.} \right),a \in \mathcal{L}_1 ,b \in \mathcal{L}_2 } \right\}$ , where $\mathcal{L}_1 ,\mathcal{L}_2 \subseteq \mathbb{F}_p^n$ for two different comprehensive codes $\mathcal{K}$. As a result we obtain two different expressions for weight enumerator of code $\mathcal{L}$.

As example we consider a case $\mathcal{L}_1 = RM_{m - 2.m} ,\mathcal{L}_2 = RM_{1.m}$, where RM _r.m is the r-th order Reed-Muller code of lenght n = 2^m.¹

Thise research was support by The Royal Swedesh akademy of Sciences