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Bounds on Box Codes

Michael Langberg, Moshe Schwartz, Itzhak Tamo

TL;DR

The paper studies box codes, a generalization of traditional $q$-ary codes where codewords have finitely many protected entries and distance is measured only on protected coordinates, with the box length defined as the average number of protected positions. It shows that $n_q^{\bx}(M,d)$ can be strictly smaller than $n_q(M,d)$ in several regimes by constructing box codes from classical codes (e.g., binary Hamming and Reed–Solomon) and inserting the unprotected symbol, and it derives asymptotic gaps such as $n^{\bx}_2(M,3) \le n_2(M,3) - 1 + o(1)$ for large $M$. The work also develops perfect box-code constructions for distances 1 and 3 using product-type and nearly-perfect 1-covering code techniques (including NP1CCs and meshing) and proves the existence of balanced, non-degenerate perfect box codes for certain lengths. Finally, it connects box codes to bipartite graph coverings, establishing lower and upper bounds on $n^{\bx}_{G,2}(M,d)$ in terms of graph parameters like the independence and chromatic numbers, thereby linking geometric box-code packing to graph-theoretic coverings with potential practical implications for energy-aware wireless communication.</n-2>

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.

Bounds on Box Codes

TL;DR

The paper studies box codes, a generalization of traditional -ary codes where codewords have finitely many protected entries and distance is measured only on protected coordinates, with the box length defined as the average number of protected positions. It shows that can be strictly smaller than in several regimes by constructing box codes from classical codes (e.g., binary Hamming and Reed–Solomon) and inserting the unprotected symbol, and it derives asymptotic gaps such as for large . The work also develops perfect box-code constructions for distances 1 and 3 using product-type and nearly-perfect 1-covering code techniques (including NP1CCs and meshing) and proves the existence of balanced, non-degenerate perfect box codes for certain lengths. Finally, it connects box codes to bipartite graph coverings, establishing lower and upper bounds on in terms of graph parameters like the independence and chromatic numbers, thereby linking geometric box-code packing to graph-theoretic coverings with potential practical implications for energy-aware wireless communication.</n-2>

Abstract

Let be the minimum length of a -ary code of size and minimum distance . Bounding is a fundamental problem that lies at the heart of coding theory. This work considers a generalization of 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 are presented.
Paper Structure (7 sections, 10 theorems, 78 equations, 1 figure)

This paper contains 7 sections, 10 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The geometric interpretation of box codes
  3. Connection to bipartite graph covering and prior results
  4. Overview
  5. A Gap Between $n_q(M,d)$ and $n^{ \mathbin{\text{ }}}_q(M,d)$
  6. Perfect Codes
  7. Covering of General Graphs

Key Result

Lemma 1

The box code $\mathcal{C}^{ \mathbin{\text{ }}}$ from Construction con:ham is well defined, for all sufficiently large $m$, and has parameters $(n^{ \mathbin{\text{ }}},M^{ \mathbin{\text{ }}},d^{ \mathbin{\text{ }}})_2$ with where $n$ and $M$ are given in eq:hamnM.

Figures (1)

  • Figure 1: An example geometric representation of a $(2,3,1)_2$ box code whose protected symbols are confined to the first $\eta=3$ locations. The code restricted to $\Sigma^3$ includes three codewords $\{(0,1,1),(1,1,{ \mathbin{\text{ }}}),({ \mathbin{\text{ }}},0,{ \mathbin{\text{ }}})\}$. $X((0,1,1))=(0,1,1)$ is represented by a gray point, $X((1,1,{ \mathbin{\text{ }}}))$ is represented by a gray line, and $X(({ \mathbin{\text{ }}},0,{ \mathbin{\text{ }}}))$ is represented by a gray box of dimension 2.

Theorems & Definitions (12)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Theorem 3
  • Example 1
  • Theorem 4
  • Corollary 1
  • Example 2
  • Theorem 5: Lower bound on $n^{ \mathbin{\text{ }}}_{G,2}(M,d)$
  • ...and 2 more