Influences of some families of error-correcting codes

TL;DR

This paper explores how the influence of individual coordinates affects the erasure recovery ability of certain error-correcting codes, introducing a new family of codes with directly computable influences.

Contribution

It introduces codes with minimum disjoint support, allowing direct influence computation, even when transitivity-based methods are not applicable.

Findings

Influences of repetition and certain weight codes are explicitly computed.

Some codes with non-transitive automorphism groups can have their influences directly determined.

Provides insight into coordinate roles in erasure repair beyond traditional symmetry assumptions.

Abstract

Binary codes of length $n$ may be viewed as subsets of vertices of the Boolean hypercube $\{0,1\}^n$. The ability of a linear error-correcting code to recover erasures is connected to influences of particular monotone Boolean functions. These functions provide insight into the role that particular coordinates play in a code's erasure repair capability. In this paper, we consider directly the influences of coordinates of a code. We describe a family of codes, called codes with minimum disjoint support, for which all influences may be determined. As a consequence, we find influences of repetition codes and certain distinct weight codes. Computing influences is typically circumvented by appealing to the transitivity of the automorphism group of the code. Some of the codes considered here fail to meet the transitivity conditions requires for these standard approaches, yet we can compute them directly.