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

2026-02-14

AI Summary

This work proposes an efficient error-correcting coding scheme for $n \times n$ arrays over a $q$-ary alphabet ($n \geq 11$, $q \geq 3$) to combat information loss caused by $(1,1)$-criss-cross deletions—simultaneous deletion of one row and one column. By combining parity checks on rows and columns with index encoding through a combinatorial construction, the scheme achieves the first explicit code whose redundancy exceeds the theoretical lower bound by only a constant term, yielding total redundancy of $2n + 2\log_q n + O(1)$, which closely approaches the optimal bound of $2n + 2\log_q n - 3$. Both encoding and decoding, including recovery from criss-cross deletions, run in $O(n^2)$ time, significantly enhancing reliability and efficiency in applications such as QR codes and DNA-based data storage.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• criss-cross deletion
• two-dimensional error-correcting codes
• redundancy-optimal
• deletion correcting codes
• array codes

Innovation

• criss-cross deletion correcting codes
• redundancy-optimal
• efficient encoding/decoding
• two-dimensional error-correcting codes
• array codes