发表机构
Indian Institute of Science, Bangalore（印度科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对含汉明、李距离的Wyner-Graham距离，推导了函数纠错码的通用Plotkin型界，其仅依赖原像集基数及集内向量成对距离和，还得到线性函数等类别的简化界，降低了界的计算复杂度。
AI 中文摘要
函数纠错码（FCCs）旨在保护消息在更高层级上的指定函数求值，而非消息本身，以此对抗错误，同时降低可靠通信所需的冗余度。迄今为止，针对匹配各类距离（包括汉明距离和李距离）的信道，人们已对FCCs展开研究。每个函数都会将消息空间划分为对应其不同函数值的原像集。现有针对所研究距离下FCC最优冗余度的Plotkin型界适用于消息空间上的任意函数，但其依赖于所有消息向量之间的成对距离，这使得这些界难以计算。我们推导了Wyner-Graham距离下FCC最优冗余度的通用Plotkin型界，该距离将汉明距离和李距离作为特例包含在内。我们的界仅依赖于原像集的基数以及每个原像集内向量之间的成对距离之和。该方法显著降低了评估现有Plotkin型界所需的计算量，并生成了更易于针对特定函数计算的简化界。我们得到了Wyner-Graham距离下线性函数的简化Plotkin型界。此外，汉明距离和李距离下线性函数的现有简化界可作为所提界的特例被推导得出。我们还得到了几类重要函数的简化界，包括汉明重量函数、汉明重量分布函数、汉明距离下的单项式函数，以及李距离下的模和函数、李重量函数和李重量分布函数。
英文摘要
Function correcting codes (FCCs) are designed to protect a specified function evaluation of messages at a higher level than the level of protection for messages, against errors while reducing the redundancy required for reliable communication. FCCs have thus far been studied for channels matched to various distances, including the Hamming and Lee distances. Every function partitions the message space into preimage sets corresponding to its distinct function values. Existing Plotkin-type bounds on the optimal redundancy of FCCs under the studied distances, applicable to arbitrary functions on the message space, depend on the pairwise distances among all the message vectors. This makes these bounds difficult to compute. We derive a general Plotkin-type bound on the optimal redundancy of FCCs under Wyner Graham distances, which include the Hamming and Lee distances as special cases. Our bound depends only on the cardinalities of the preimage sets and the sum of pairwise distances only among vectors within each preimage set. This approach significantly reduces the computations required to evaluate the existing Plotkin-type bounds and yields simplified bounds that are easier to compute for specific functions. We obtain simplified Plotkin-type bound for linear functions under the Wyner-Graham distance. Furthermore, the existing simplified bounds for linear functions under the Hamming and Lee distances are recovered as special cases of the proposed bound. We also obtain simplified bounds for several important classes of functions, including the Hamming weight function, the Hamming weight distribution function, the monomial functions under the Hamming distance, and the modular sum function, the Lee weight function, and the Lee weight distribution function under the Lee distance.
Comments22 pages; Submitted to a journal
