Generalized Function-Correcting Partition Codes: Bounds and constructions
University of Bristol Mumbai Enterprise Campus（布里斯托大学孟买企业校区）；International Institute of Information Technology Hyderabad（印度信息技术研究所海德拉巴分校）
Charul Rajput, Mahak, V. Lalitha
备注：
We derived linear programming bounds on the optimal redundancy. A shorter version of this work has been accepted for presentation at the IEEE Information Theory Workshop (ITW), 2026
We introduce generalized function-correcting partition codes (GFCPCs) that simultaneously protect multiple partitions of the message space against different numbers of errors. Given partitions with respective distance requirements, a GFCPC is a systematic encoding that guarantees, for each partition, a specified minimum Hamming distance between codewords whose messages lie in different blocks. This framework unifies and generalizes both function-correcting partition codes, which protect multiple functions with a common error-correction level, and function-correcting codes with data protection, which assign different levels of protection to data and a single function. We define the distance requirement matrix $\mathcal{D}$ for GFCPCs and use it to characterize the optimal redundancy in terms of the shortest length of an associated $\mathcal{D}$-code. We derive general upper and lower bounds on the optimal redundancy, including an upper bound that considers the join of different combinations of the partitions. For two partitions of the message space over the binary field, we establish improved lower bounds on the optimal redundancy under specific neighborhood conditions on the partitions. We develop a linear programming bound on the optimal redundancy of GFCPCs. The bound specializes to function-correcting codes and function-correcting codes with data protection, for which, to the best of our knowledge, no such bound was previously known. We present a multi-step construction procedure for these codes and illustrate it with some examples. Through several examples, we demonstrate that the proposed framework can yield strictly smaller redundancy than both the sum of the individual FCPC redundancies and the redundancy of a single FCPC designed for the join partition with the highest distance (strongest protection required).
Error-Correcting Functional Index Codes, Generalized Exclusive Laws and Graph Coloring
我们考虑\emph{functional index coding problem}在一个无差错的广播网络中，源产生一组消息，有多个接收器，每个持有一组功能的源消息在其缓存，称为\emph{Has-set}，并要求知道另一组功能的消息，称为\emph{Want-set}。认识到接收器的\emph{Has-sets}，源旨在通过进行编码传输（称为\emph{functional index code}）来满足每个接收器的需求。目的是尽量减少所需的这种传输次数。接收器的需求对代码的限制通过称为\emph{generalized exclusive law}的约束来表示，并且使用利用这些约束构造的\emph{confusion graph}来获得代码。基于混淆图的参数的最佳代码的大小的界限。接下来，我们考虑的情况下，错误的传输，并提供一个必要和充分的条件，FIC必须满足在每个接收器的正确解码所需的功能，并获得一个纠错FIC的长度上的下界。
Entropic proofs of Singleton bounds for quantum error-correcting codes
我们表明，一个相对简单的推理，使用冯诺伊曼熵不等式产生一个强大的证明量子单态界的量子纠错码（QECC）。对于纠缠辅助量子纠错码（EAQECC）和催化码（CQECC），一种类型的广义量子单例界[Brun等人，IEEE Trans. Inf. Theory 60（6）：3073--3089（2014）]被认为成立多年，直到最近我们中的一个人发现了一个反例[MG，Phys.Rev.A103，020601（2021）]。在这里，我们纠正这种状况，通过证明正确的广义量子单例界，扩展上述证明方法QECC，我们还证明了信息理论上的紧界的纠缠通信权衡EAQECC。所有的界限相关的块长度$n$和代码长度$k$为给定的最小距离$d$，我们表明，他们是强大的，在这个意义上说，他们持有的代码，只纠正大部分的擦除错误小于$d$字母的小扰动。与经典的情况相比，边界采取定性不同的形式，这取决于最小距离是否小于或大于块长度的一半。我们还提供了一个传播规则：任何纯QECC产生EAQECC具有相同的距离和尺寸，但更短的块长度。
