AI 中文总结
本文研究源自t-设计、差集、BIBD等组合结构的循环码、BIBD-LDPC码等线性码类的数据符号最大可达服务率下界，推导了相关码的下界及Singer差集循环码的精确值。
AI 中文摘要
本文研究特定线性码类（包括循环码和低密度奇偶校验（LDPC）码）中数据符号的最大可达服务率的下界，这些线性码源自组合结构，如t-设计、差集和平衡不完全区组设计（BIBD）。我们首先针对以下两种情况，为每个数据符号的最大可达服务率建立下界：(i) 假设C⊥中固定重量码字的支撑集构成BIBD的系统线性码C；(ii) 假设C⊥中固定重量码字的支撑集构成t-设计的非系统二元码。随后，我们研究由BIBD关联矩阵得到的线性码，特别是某些类BIBD-LDPC码，展示如何利用底层设计的参数确定对应线性码数据符号最大可达服务率的下界。此外，我们分析系统扩展线性码的最大可达服务率，证明存在与对偶码字对应的对称BIBD（SBIBD）可用于推导关联系统循环码的最大可达服务率下界。我们还给出若干由差集构造的循环码族，并得到其数据符号最大可达服务率的显式下界，尤其确定了源自Singer差集的循环码每个数据符号的最大可达服务率的精确值。
英文摘要
In this paper, we investigate lower bounds on the maximum achievable service rates for data symbols in certain classes of linear codes, including cyclic codes and low-density parity-check (LDPC) codes, that are derived from combinatorial structures such as $t$-designs, difference sets, and balanced incomplete block designs (BIBDs). We first establish a lower bound on the maximum achievable service rate for each data symbol in the following two cases: (i) systematic linear codes $C$ under the assumption that the supports of codewords of a fixed weight in $C^\perp$ form a BIBD, and (ii) non-systematic binary codes under the assumption that the supports of codewords of a fixed weight in $C^\perp$ form a $t$-design. We then investigate the linear codes obtained from the incidence matrices of BIBDs, particularly certain classes of BIBD-LDPC codes, and show how the parameters of the underlying designs can be exploited to determine lower bounds on the maximum achievable service rates for the data symbols of the corresponding linear code. In addition, we analyze the maximum achievable service rates of systematic extended linear codes. We show that the existence of a symmetric BIBD (SBIBD) corresponding to a dual codeword can be used to derive a lower bound on the maximum achievable service rate of the associated systematic cyclic code. We also present some families of cyclic codes constructed from difference sets and obtain explicit lower bounds on the maximum achievable service rates for their data symbols. In particular, we determine the exact values of the maximum achievable service rates for each data symbol of cyclic codes arising from Singer difference sets.
