Alphabet Size Matching Techniques Based on Non-Binary Gilbert-Varshamov Bounded Limits for Synchronization Finite State Markov Channel

Abstract

The Gilbert-Varshamov (GV) lower bound is used to provide indications and prescriptions for the outer code coding parameters for a memory synchronisation model that focuses solely on the internal resynchronisation process. The binary and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary GV bounds are utilised in this analysis to indicate parameters to remove the remaining substitution errors and provide a complete framework. Procedures and examples are provided to determine optimal outer code parameters for given inner-entropies and residual substitution errors produced during resynchronisation. In particular, using the non-binary GV bounds allows us to match the best alphabet size for given parameters. For the cases explored, a 16-ary GV bound provides the best results, with an ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> ) code of (120, 57, 37) being a possible outer code when the inner entropy is 0.1. Using GV bounds for outer code parameter considerations frees the system from using stringent codes and instead allows any outer code to be utilised to meet the required error correction needs.