发表机构
Indian Institute of Science（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
本文针对符号对读取信道，将函数校正码推广至李度量，定义函数校正符号对李距离码，并推导其最优冗余的Plotkin型与Gilbert-Varshamov型界，给出多种函数的构造与界。
AI 中文摘要
函数校正码（FCCs）保护消息的指定函数求值免受错误影响，同时减少可靠通信所需的冗余。我们研究符号对读取信道的FCCs，其动机来自读取重叠符号对的高密度存储系统。相移键控（PSK）调制因其带宽效率和噪声鲁棒性而非常适合此类系统。虽然符号对读取信道的FCCs已在汉明度量下得到研究，但李度量是$q$进制PSK更合适的错误模型，因此也适用于$q$进制符号对读取信道。我们将符号对李距离（先前仅定义在$\mathbb{Z}_4$上）推广到$\mathbb{Z}_q$（$q\ge2$），并引入定义在$\mathbb{Z}_q$上的函数校正符号对李距离码（FCSPLCs），专门针对$2^m$进制PSK星座取$q=2^m$（$m\ge1$）。我们通过引入不规则对李距离码并将其最优冗余与这类码的最短长度相关联，研究其冗余需求。我们通过不规则对李距离码最短长度的下界和上界，推导出最优冗余的Plotkin型和Gilbert--Varshamov型界。对于双射函数，我们获得$\mathbb{Z}_q$上符号对读取信道经典李度量码的相应Plotkin型和Gilbert--Varshamov型界，据我们所知，这是李度量符号对设置中首次出现此类界。然后，我们将FCSPLC框架专门应用于对局部有界函数、对李权重函数和对李权重分布函数，给出显式构造和相应的最优冗余界。最后，对于线性函数，我们推导出最优冗余的Plotkin型下界。
英文摘要
Function-correcting codes (FCCs) protect specified function evaluations of messages against errors while reducing the redundancy required for reliable communication. We study FCCs for symbol-pair read channels, motivated by high-density storage systems that read overlapping symbol pairs. Phase-shift keying (PSK) modulation is well-suited to such systems due to its bandwidth efficiency and noise robustness. While FCCs for symbol-pair read channels have been studied under the Hamming metric, the Lee metric is a more appropriate error model for $q$-ary PSK and, hence for $q$-ary symbol-pair read channels. We generalize the symbol-pair Lee distance, previously defined only over $\mathbb{Z}_4$, to $\mathbb{Z}_q$, $q\ge2$, and introduce function-correcting symbol-pair Lee-distance codes (FCSPLCs) over $\mathbb{Z}_q$, specializing to $q=2^m$, $m\ge1$, for $2^m$-ary PSK constellations. We investigate their redundancy requirements by introducing irregular-pair Lee-distance codes and relating the optimal redundancy of FCSPLCs to the shortest length of such codes. We derive Plotkin-type and Gilbert--Varshamov-type bounds on the optimal redundancy through lower and upper bounds on the shortest length of irregular-pair Lee-distance codes. For bijective functions, we obtain corresponding Plotkin-type and Gilbert--Varshamov-type bounds for classical Lee metric codes for symbol-pair read channels over $\mathbb{Z}_q$, which, to the best of our knowledge, are the first such bounds for the Lee metric symbol-pair setting. We then specialize the FCSPLC framework to pair-locally bounded functions, the Pair-Lee weight function, and the Pair-Lee weight distribution function, giving explicit constructions and corresponding bounds on the optimal redundancy. Finally, for linear functions, we derive a Plotkin-type lower bound on the optimal redundancy.
Comments27 pages; A short version to be communicated to ITW27
