This paper investigates the problem of correcting multiple criss-cross insertions and deletions in arrays. More precisely, we study the unique recovery of $n \times n$ arrays affected by $t$-criss-cross deletions defined as any combination of $t_r$ row and $t_c$ column deletions such that $t_r + t_c = t$ for a given $t$. We show an equivalence between correcting $t$-criss-cross deletions and $t$-criss-cross insertions and show that a code correcting $t$-criss-cross insertions/deletions has redundancy at least $tn + t \log n - \log(t!)$. Then, we present an existential construction of $t$-criss-cross insertion/deletion correcting code with redundancy bounded from above by $tn + \mathcal{O}(t^2 \log^2 n)$. The main ingredients of the presented code construction are systematic binary $t$-deletion correcting codes and Gabidulin codes. The first ingredient helps locating the indices of the inserted/deleted rows and columns, thus transforming the insertion/deletion-correction problem into a row/column erasure-correction problem which is then solved using the second ingredient.
翻译:本文调查了纠正多个 crits- cross 插入和在阵列中删除的问题。 更准确地说, 我们研究受美元- cross- cross- deletment 影响、 美元- cross- delete Expressions expressions $tn + t_ r$ r美元行和 $t_ c 列删除的任何组合, 使美元+ t_ c = t美元 = 美元- t美元 = 美元- c = 美元( 美元- cross- cross- cross- cross 插入) 和 美元- 美元- cross- cross- cross explexptions (美元- 美元- 美元- 美元- 美元- cross- cross- crosplemental expilations) 和 rodefinal- drodeal- rodestrual- dromaism- romaisl- preflegisl- preal- preal- codestrisl- 和 lifil- 美元/ drefol- cregilmental- 美元/ clifol- roclemental- rogild- 和 美元/ cligilmental- 要求- cligilmental- 要求/ clicol- 要求, 然后译/ 平序/ 列/ 和 平序/ 平列/ 平序/ 平列/ 平列/ 度/ 平列/ 平序/ 度/ 平列/ 度/ 平列/ 度/ 度/ 度/ 度/ 度/ 平序/ 度/ 度/ 度/ 度/ 度- 度/ 等等) 问题。。。 问题。 问题, 然后) 问题。。 和 和 将 将 将 和 将 将 将 将 将 将 和 将 将 将 将 将 将 将 将 将 变正正正正正正正正