A Fresh Look at Coding for q-ary Symmetric Channels

This paper studies coding schemes for the $q$-ary symmetric channel based on binary low-density parity-check (LDPC) codes that work for any alphabet size $q=2^m$, $m\in\mathbb{N}$, thus complementing some recently proposed packet-based schemes requiring large $q$. First, theoretical optimality of a simple layered scheme is shown, then a practical coding scheme based on a simple modification of standard binary LDPC decoding is proposed. The decoder is derived from first principles and using a factor-graph representation of a front-end that maps $q$-ary symbols to groups of $m$ bits connected to a binary code. The front-end can be processed with a complexity that is linear in $m=\log_2 q$. An extrinsic information transfer chart analysis is carried out and used for code optimization. Finally, it is shown how the same decoder structure can also be applied to a larger class of $q$-ary channels.