发表机构
Nanyang Technological University; Xi’an Jiaotong University（南洋理工大学; 西安交通大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究固定列表大小的插入-删除码，通过组合约简和Lovász局部引理推导新可达速率界，严格改进Levenshtein下界，并给出上界。
AI 中文摘要
我们研究对抗性插入和删除的纠错码，其列表大小$L$独立于码长固定。我们推导了二进制码的新可达速率界，以及每个固定字母表大小$q\ge2$上的上界，并保留对$L$的显式依赖。我们建立了一个组合约简，在解码保证中将$L$单位的插入预算换为1单位的删除预算，而不改变码或增加列表大小。因此，混合错误（插入比例为$\gamma$，删除比例为$\delta$）的渐近界可由仅插入下界（在$\gamma+L\delta$处）和仅删除上界（在$\delta+\gamma/L$处）得出。对于二进制唯一解码，我们严格改进了Levenshtein的经典渐近速率下界，对于每个删除比例$0<\delta<1/2$且经典速率表达式非负的情况。在$\delta=0.1$时，下界从约$0.162009$增加到$0.180431$，相对增加约$11.37\\%$。我们的框架还为每个固定列表大小提供了插入和删除下界。存在性证明结合了Lovász局部引理与从具有指定游程数的单词中采样，其中游程是相等符号的最大块。生成函数提供了$L+1$个采样单词共享一个允许接收单词的概率的精细界。我们还通过游程计数推导了一个Levenshtein型上界，并分别通过用于将$L+1$个码字嵌入公共超序列的位置集的交集和并集推导了一个高阶Elias界。后者在$L=1$时恢复了Yasunaga的渐近唯一解码界，并严格改进了Haeupler--Shahrasbi--Sudan插入界，对于每个固定$L$和$0<\gamma<q-1$。数值比较量化了增益和剩余差距。
英文摘要
We study codes correcting adversarial insertions and deletions with list size $L$ fixed independently of the block length. We derive new achievable-rate bounds for binary codes and upper bounds over every fixed alphabet of size $q\ge2$, retaining explicit dependence on $L$. We establish a combinatorial reduction that trades $L$ units of insertion budget for one unit of deletion budget in the decoding guarantee, without changing the code or increasing the list size. Consequently, asymptotic bounds for mixed errors with insertion fraction $γ$ and deletion fraction $δ$ follow from insertion-only lower bounds at $γ+Lδ$ and deletion-only upper bounds at $δ+γ/L$. For binary unique decoding, we strictly improve Levenshtein's classical asymptotic rate lower bound for every deletion fraction $0<δ<1/2$ for which the classical rate expression is nonnegative. At $δ=0.1$, the lower bound increases from approximately $0.162009$ to $0.180431$, a relative increase of about $11.37\%$. Our framework also yields insertion and deletion lower bounds for every fixed list size. The existence proofs combine the Lovász local lemma with sampling from words having a specified number of runs, where a run is a maximal block of equal symbols. Generating functions provide refined bounds on the probability that $L+1$ sampled words share an allowed received word. We also derive a Levenshtein-type upper bound by run counting and, separately, a higher-order Elias bound using intersections and unions of the position sets used to embed $L+1$ codewords in a common supersequence. The latter recovers Yasunaga's asymptotic unique-decoding bound at $L=1$ and strictly improves the Haeupler--Shahrasbi--Sudan insertion bound for every fixed $L$ and $0<γ<q-1$. Numerical comparisons quantify the gains and remaining gaps.
Comments46 pages, 5 figures
