Influences of some families of error-correcting codes
Hailey Egan, Jason T. LeGrow, Gretchen L. Matthews, Jeff Suliga
TL;DR
This work links erasure recovery in binary codes to the influences of coordinates of monotone Boolean functions, introducing minimum disjoint support codes as a tractable class for direct influence computation. It formalizes the influence framework with sets $\Omega_i$, $S_i$, and the $j$-boundary $B_j$, and derives exact $I_j^{(p)}$ expressions for several code families. The results show that simple parity-check and repetition-like codes exhibit uniform coordinate influences, while distinct weight codes and hybrids can have highly varying influences depending on their partition structure. Although the studied families have rates tending to zero, the approach provides structural insights into how code symmetry and support organization affect erasure-recovery capabilities, and suggests directions for identifying new capacity‑achieving families via influence analysis.
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.