🤖 AI Summary
This work addresses the limitation of classical weight distributions in linear codes, which fail to capture coordinate-wise nonzero activity with sufficient granularity. To overcome this, the paper introduces, for the first time, the notions of support distribution and support enumerator, refining the traditional weight distribution to enable a coordinate-level characterization of nonzero symbol occurrences. The authors establish explicit counting formulas for such coordinate-wise activity and derive a MacWilliams-type identity relating the support enumerators of a linear code and its dual. Building on this framework, they formulate an algebraic duality relation for support enumerators and provide a novel criterion for self-dual codes based on the equality of their support distributions. This contribution deepens the structural understanding of linear codes, enriches duality theory at the coordinate level, and offers a new tool for constructing and identifying self-dual codes.
📝 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.