AI 中文总结
本文研究$(t,e)$-复合非对称纠错码的构造，提出新构造方法，给出最优$(2,e)$-CAECCs，引入弱$B_e$-集，还分析了列表解码放宽要求后的相关结果。
AI 中文摘要
本文聚焦于近期研究的$(t,e)$-复合非对称纠错码（$(t,e)$-CAECCs）的唯一可译码/列表可译码的构造。设$X$为$m\times n$的二元矩阵，其中每行的汉明重量为$w$。当$X$中至多$t$行出现错误，且这些出错行中每行至多有$e$个$1\to0$错误时，称$X$中发生了$(t,e)$-复合非对称错误。针对一般的$m,n,w,t,e$，我们提出了冗余度至多为$(t-1)\text{log}(m)+O(1)$的$(t,e)$-CAECCs的新构造，其中$O(1)$是与码长$m$无关的常数。特别地，这给出了一类在冗余度方面最优的$(2,e)$-CAECCs（将冗余度视为行数$m$的函数）。当$m$为素数幂时，冗余度可进一步降至$(t-1)\text{log}(m)-O(\text{log}(m))$。为进一步增大这些码的规模，我们引入了一种名为弱$B_e$-集的组合对象。当$e=w$时，我们展示了对所提码进行高效编码/解码的方法。最后，我们研究若将唯一解码的要求放宽为列表解码可获得的增益，结果表明当列表大小为$t!$或$t$的指数函数时，存在具有恒定冗余度的列表可译码$(t,e)$-CAECCs；当列表大小为2时，存在冗余度为$\text{log}(m)+O(1)$的列表可译码$(3,2)$-CAECCs。
英文摘要
In this paper, we focus on constructions of unique-decodable/list-decodable on the recently studied $(t,e)$-composite-asymmetric error-correcting codes ($(t,e)$-CAECCs). Let $X$ be an $m\times n$ binary matrix, in which each row has Hamming weight $w$. When at most $t$ rows of $X$ suffer from errors and in each of these erroneous rows, there are at most $e$ $1 \to 0$ errors, we say that a $(t,e)$-composite-asymmetric-error occurs in $X$. For general $m,n,w,t,e$, we propose new constructions of $(t,e)$-CAECCs with redundancy at most $(t-1)\log(m)+O(1)$, where $O(1)$ is a number independent of the code-length $m$. In particular, this gives a class of $(2,e)$-CAECCs that are optimal in terms of their redundancy. %(in terms of redundancy, regarded as a function of the number of rows $m$) s. When $m$ is a prime power, the redundancy can be further reduced to $(t-1)\log(m)-O(\log(m))$. To further increase the size of these codes, we introduce a combinatorial object called a weak $B_e$-sets. When $e=w$, we show an efficient way to encode/decode our codes. At last, we investigate how much we can gain if we relax the requirement of uniquely decoding to list-decoding. It is shown that when the list size is $t!$ or an exponential function of $t$, there are list-decodable $(t,e)$-CAECCs with constant redundancy. When the list size is two, we show that there are list-decodable $(3,2)$-CAECCs with redundancy $\log(m)+O(1)$.
