On Multidimensional 2-Weight-Limited Burst-Correcting Codes

TL;DR

The paper addresses multidimensional burst-correcting codes for bursts of weight at most $2$ under three models: $L_\infty$, $L_1$, and straight axis-parallel bursts. It develops explicit linear constructions using parity-check matrices built from BCH codes (including a Lee-metric BCH for the $L_1$ model) and packing designs for the straight model, accompanied by lower bounds on excess redundancy. The main contributions are concrete decoding procedures and redundancy analyses for all three models, along with extended-length variants to reduce redundancy and a comparative table of performance. Overall, the work advances practical, dimension-aware burst-correcting codes for multidimensional storage and communication, achieving near-optimal redundancy in several regimes and clarifying trade-offs across burst-shape models.

Abstract

We consider multidimensional codes capable of correcting a burst error of weight at most $2$. When two positions are in error, the burst limits their relative position. We study three such limitations: the $L_\infty$ distance between the positions is bounded, the $L_1$ distance between the positions is bounded, or the two positions are on an axis-parallel line with bounded distance between them. In all cases we provide explicit code constructions, and compare their excess redundancy to a lower bound we prove.

Figures (1)

• Figure 1: A depiction of the three $b$-closeness models. Identify each square with a point of $\mathbb{Z}^2$, and let $\mathbf{i}$ be the shaded square. The $2$-dimensional $3$-close positions to $\mathbf{i}$ are then the squares in (a) the $L_\infty$ model, (b) the $L_1$ model, and (c) the straight model.

Theorems & Definitions (31)

• Definition 1
• Proposition 1
• proof
• Theorem 2
• proof
• Corollary 3
• proof
• Theorem 4
• proof
• Corollary 5