High-rate LDPC codes from partially balanced incomplete block designs

Abstract

This paper presents a combinatorial construction of low-density parity-check (LDPC) codes from partially balanced incomplete block designs. Since Gallager's construction of LDPC codes by randomly allocating bits in a sparse parity-check matrix, many researchers have used a variety of more structured combinatorial approaches. Many of these constructions start with the Galois field; however, this limits the choice of parameters of the constructed codes. Here we present a construction of LDPC codes of length 4n(2) - 2n for all n using the cyclic group of order 2n. These codes achieve high information rate (greater than 0.8) for n >= 8, have girth at least 6 and have minimum distance 6 for n odd. The results provide proof of concept and lay the groundwork for potential high performing codes.