Bounds on Box Codes

TL;DR

This paper introduces bounds on a generalized version of code length for box codes, which are codes with protected and unprotected entries, extending classical bounds in coding theory.

Contribution

It provides the first bounds on the length of box codes, a generalization involving protected entries, expanding understanding of code structures with partial protection.

Findings

Derived upper bounds on box code lengths

Established lower bounds for box code parameters

Extended classical coding bounds to the box code setting

Abstract

Let $n_q(M,d)$ be the minimum length of a $q$-ary code of size $M$ and minimum distance $d$. Bounding $n_q(M,d)$ is a fundamental problem that lies at the heart of coding theory. This work considers a generalization $n^\bx_q(M,d)$ of $n_q(M,d)$ corresponding to codes in which codewords have \emph{protected} and \emph{unprotected} entries; where (analogs of) distance and of length are measured with respect to protected entries only. Such codes, here referred to as \emph{box codes}, have seen prior studies in the context of bipartite graph covering. Upper and lower bounds on $n^\bx_q(M,d)$ are presented.