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

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 $q=\mathrm{\Omega }(n)$ (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.