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 $q = Ω(n)$ (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.

翻译：二维纠错码，其码字表示为$q$元字母表上的$n \times n$阵列，在QR码、DNA存储和赛道存储器等领域具有重要应用。在可能的错误模式中，$(t_r,t_c)$-交叉删除——即同时删除$t_r$行和$t_c$列——具有特殊的重要性。本文聚焦于$q$元$(1,1)$-交叉删除纠错码。我们提出了一种新颖的码构造，并为参数$n \ge 11$和$q \ge 3$开发了完整的编码、解码和数据恢复算法。所提编码、解码和数据恢复算法的复杂度为$\mathcal{O}(n^2)$。此外，我们证明对于$n \ge 11$且$q = Ω(n)$（即存在常数$c>0$使得$q \ge cn$），所构造码的码冗余度和编码器冗余度均为$2n + 2\log_q n + \mathcal{O}(1)$，这在下界（$2n + 2\log_q n - 3$）的$\mathcal{O}(1)$间隙内达到了最优。据我们所知，这是首个能以仅$\mathcal{O}(1)$的间隙达到最优冗余度，同时具备显式编码和解码算法的构造。