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

TL;DR

This work develops a theory for two-dimensional criss-cross deletions in $n\times n$ arrays over a $q$-ary alphabet, establishing a sphere-packing lower bound and a Gilbert-Varshamov upper bound on code redundancy and showing the optimal redundancy lies within a tight range of $(t_r+t_c)n\log q+\Theta(\log n)$. It then provides near-optimal and optimal constructions for the pivotal $(1,1)$ case: a non-binary construction and an arbitrary-alphabet construction, both achieving redundancy $2n\log q+2\log n+O(1)$ (up to constants) with $O(n^2)$ decoding time. The paper also extends these ideas to $(t_r,t_c)$-criss-cross burst deletions, delivering burst-correcting codes with matching optimal redundancy up to additive constants. Collectively, these results advance both the theoretical bounds and practical decoding schemes for two-dimensional deletion correction, with potential impact on data storage and imaging applications.

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 $Σ_q := \{0, 1, \ldots, q-1\}$ 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 $\mathcal{C} \subseteq Σ_q^{n \times n}$ 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.

Figures (1)

• Figure 1: Illustration of the criss-cross deletion process. Assume that $\boldsymbol{Y}$ is obtained form $\boldsymbol{X}$ by deleting its $i$-th row and $j$-th column. We can visualize the process of obtaining $\boldsymbol{Y}$ as follows: First, we move the $i$-th row of $\boldsymbol{X}$ to the bottom, resulting in the array $\boldsymbol{X}'$. Next, we shift the $j$-th column of $\boldsymbol{X}'$ to the rightmost position, yielding $\boldsymbol{X}"$. Finally, $\boldsymbol{Y}$ is the top-left corner subarray of $\boldsymbol{X}"$, with size $(n-1) \times (n-1)$.

Theorems & Definitions (49)

• Remark 1
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Remark 2
• Definition 1
• Lemma 5
• Corollary 1
• Lemma 6