Redundancy-Optimal Constructions of $(1,1)$-Criss-Cross Deletion Correcting Codes with Efficient Encoding/Decoding Algorithms

TL;DR

This paper introduces a new construction for $(1,1)$-criss-cross deletion correcting codes over $q$-ary alphabets, achieving near-optimal redundancy with efficient encoding and decoding algorithms for array sizes $n \\ge 11$ and $q \\ge 3$.

Contribution

The paper presents the first explicit construction of $(1,1)$-criss-cross deletion correcting codes with redundancy close to the theoretical lower bound and efficient algorithms.

Findings

Redundancy of $2n + 2\\log_q n + \\mathcal{O}(1)$ achieved.

Encoding, decoding, and data recovery algorithms operate in $\\mathcal{O}(n^2)$ time.

Construction is valid for $n \\ge 11$ and $q \\ge 3$, with redundancy optimal within an $\\mathcal{O}(1)$ gap.

Abstract

Two-dimensional error-correcting codes, where codewords are represented as $n \times n$ arrays over a $q$-ary alphabet, find important applications in areas such as QR codes, DNA-based storage, and racetrack memories. Among the possible error patterns, $(t_r,t_c)$-criss-cross deletions-where $t_r$ rows and $t_c$ columns are simultaneously deleted-are of particular significance. In this paper, we focus on $q$-ary $(1,1)$-criss-cross deletion correcting codes. We present a novel code construction and develop complete encoding, decoding, and data recovery algorithms for parameters $n \ge 11$ and $q \ge 3$. The complexity of the proposed encoding, decoding, and data recovery algorithms is $\mathcal{O}(n^2)$. Furthermore, we show that for $n \ge 11$ and $q = \Omega(n)$ (i.e., there exists a constant $c>0$ such that $q \ge cn$), both the code redundancy and the encoder redundancy of the constructed codes are $2n + 2\log_q n + \mathcal{O}(1)$, which attain the lower bound ($2n + 2\log_q n - 3$) within an $\mathcal{O}(1)$ gap. To the best of our knowledge, this is the first construction that can achieve the optimal redundancy with only an $\mathcal{O}(1)$ gap, while simultaneously featuring explicit encoding and decoding algorithms.