Non-Binary LDPC Erasure Codes With.pdf

IEEE TRANSACTIONS MUNICATIONS, VOL. 63, NO. 11, NOVEMBER 2015 3937 Non-Binary LDPC Erasure Codes With Separated Low-Degree Variable Nodes Giuliano Garrammone, Member, IEEE , Enrico Paolini, Member, IEEE , Balázs Matuz, Member, IEEE ,and Gianluigi Liva, Senior Member, IEEE Abstract —The code design of non-binary low-density parity- check codes for the erasure channel, under maximuma posteriori decoding, is addressed. In particular, a partially structured en- semble of codes, characterized by a careful control of the amount and of the connectivity of the variable nodes of small degrees, is proposed. The identi?ed ensemble of codes is analyzed in terms of asymptotic thresholds and weight distribution and it is shown that codes from the ensemble provide a remarkable trade-off between waterfall performance, error ?oor, and plexity. As an example, the performance curve of a short (256,128) code on the memoryless 16-ary erasure channel tightly approaches the Singleton bound at least down to a codeword error rate of 10 ? 9 , at low plexity. Index Terms —Erasure channel, ?nite ?elds, low-density parity- check codes, maximuma posteriori decoding. I. I NTRODUCTION A S SIMPLE models of munication channels, erasure channels [1] have attr acted an increasing attention in the past years and meanwhile th e design of erasure correcting codes approaching the Shannon limit has been the subject of several investigations [2]–[7]. Remarkably, it was proved that binary low-density parity-check (LDPC) codes [8] under belief propagation (BP) decoding are able to achieve the Shannon capacity of the binary erasure ch annel (BEC) when their code- word length tends to in?nity [5], even under constraints on their plexity [9], and that binary Raptor codes are universally capacity achieving over the BEC with a plexity that is linear in the number of input symbols [6]. Although they are able to achieve the BEC capacity, in the practical ?nite-length regime binary LDPC codes under BP decoding exhib