On the Algebraic Structure Underlying the Support Enumerators of Linear Codes

TL;DR

This paper introduces support distribution and support enumerator concepts for linear codes, providing formulas and identities that deepen understanding of code structure and duality at the coordinate level.

Contribution

It develops support enumerator concepts, establishes formulas for coordinate activity, and derives a MacWilliam's type identity relating a code and its dual.

Findings

Derived formulas for counting codewords with nonzero coordinates

Established a MacWilliam's type identity for support enumerators

Provided a condition for self-duality based on support distributions

Abstract

In this paper, we have introduced the concepts of support distribution and the support enumerator as refinements of the classical weight distribution and weight enumerator respectively, capturing coordinate level activity in linear block codes. More precisely, we have established formula for counting codewords in the linear code C whose i-th coordinate is nonzero. Moreover, we derived a MacWilliam's type identity, relating the normalized support enumerators of a linear code and its dual, explaining how coordinate information transforms under duality. Using this identity we deduce a condition for self duality based on the equality of support distributions. These results provide a more detailed understanding of code structure and complement classical weight based duality theory.