Maximal achievable service rates of some classes of linear codes

Abstract

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.