Criss-Cross Deletion Correcting Codes: Optimal Constructions with Efficient Decoders

TL;DR

This paper develops optimal two-dimensional error-correcting codes for criss-cross deletions in arrays, providing bounds, explicit constructions for certain cases, and efficient decoding algorithms, advancing the theory and practice of error correction in 2D data.

Contribution

It introduces the first optimal constructions and decoding algorithms for criss-cross deletion correcting codes in two dimensions.

Findings

Derived bounds on code redundancy for criss-cross deletions.

Proposed optimal code constructions for (1,1)-criss-cross deletions.

Developed efficient $O(n^2)$ decoding algorithms for the codes.

Abstract

This paper addresses fundamental challenges in two-dimensional error correction by constructing optimal codes for \emph{criss-cross deletions}. We consider an $ n \times n $ array $\boldsymbol{X}$ over a $ q $-ary alphabet $\Sigma_q := \{0, 1, \ldots, q-1\}$ that is subject to a \emph{$(t_r, t_c)$-criss-cross deletion}, which involves the simultaneous removal of $ t_r $ rows and $ t_c $ columns. A code $\mathcal{C} \subseteq \Sigma_q^{n \times n}$ is defined as a \emph{$(t_r,t_c)$-criss-cross deletion correcting code} if it can successfully correct these deletions. We derive a sphere-packing type lower bound and a Gilbert-Varshamov type upper bound on the redundancy of optimal codes. Our results indicate that the optimal redundancy for a $(t_r, t_c)$-criss-cross deletion correcting code lies between $(t_r + t_c)n\log q + (t_r + t_c)\log n + O_{q,t_r,t_c}(1)$ and $(t_r + t_c)n\log q + 2(t_r + t_c)\log n + O_{q,t_r,t_c}(1)$, where the logarithm is on base two, and $O_{q,t_r,t_c}(1)$ is a constant that depends solely on $q$, $t_r$, and $t_c$. For the case of $(1,1)$-criss-cross deletions, we propose two families of constructions that achieve $2n\log q + 2\log n + O_q(1)$ bits of redundancy. This redundancy is optimal up to an additive constant term $O_q(1)$, which depends solely on $q$. One family is designed for non-binary alphabets, while the other addresses arbitrary alphabets. For the case of $(t_r, t_c)$-criss-cross deletions, we provide a strategy to derive optimal codes when both unidirectional deletions occur consecutively. We propose decoding algorithms with a time complexity of $O(n^2)$ for our codes, which are optimal for two-dimensional scenarios.