An Algebraic Framework for Concatenated Linear Block Codes in Side Information Based Problems

TL;DR

This paper develops an algebraic framework for concatenated linear block codes in side information problems, demonstrating that nested properties are preserved and enabling new code designs for source and channel coding.

Contribution

It introduces a novel algebraic approach showing that concatenated codes can maintain nested properties, expanding possibilities for code combinations in side information scenarios.

Findings

Concatenated codes can achieve rate-distortion and capacity-noise bounds.

Nested properties are preserved through concatenation, even with only one nested code.

Binary inner codes combined with non-binary outer codes address practical problems.

Abstract

This work provides an algebraic framework for source coding with decoder side information and its dual problem, channel coding with encoder side information, showing that nested concatenated codes can achieve the corresponding rate-distortion and capacity-noise bounds. We show that code concatenation preserves the nested properties of codes and that only one of the concatenated codes needs to be nested, which opens up a wide range of possible new code combinations for these side information based problems. In particular, the practically important binary version of these problems can be addressed by concatenating binary inner and non-binary outer linear codes. By observing that list decoding with folded Reed- Solomon codes is asymptotically optimal for encoding IID q-ary sources and that in concatenation with inner binary codes it can asymptotically achieve the rate-distortion bound for a Bernoulli symmetric source, we illustrate our findings with a new algebraic construction which comprises concatenated nested cyclic codes and binary linear block codes.