Bounds on the decoding complexity of punctured codes on graphs

TL;DR

This paper introduces new sequences of irregular repeat-accumulate codes that achieve capacity on the BEC with bounded decoding complexity, contrasting with previous codes whose complexity grows with the inverse of the gap to capacity.

Contribution

The authors propose punctured irregular repeat-accumulate codes that maintain bounded complexity while achieving capacity, and derive a universal lower bound on decoding complexity for punctured codes.

Findings

Capacity-achieving codes with bounded complexity on BEC

Puncturing enables sufficient state nodes in Tanner graphs

Universal lower bound on decoding complexity for punctured codes

Abstract

We present two sequences of ensembles of non-systematic irregular repeat-accumulate codes which asymptotically (as their block length tends to infinity) achieve capacity on the binary erasure channel (BEC) with bounded complexity per information bit. This is in contrast to all previous constructions of capacity-achieving sequences of ensembles whose complexity grows at least like the log of the inverse of the gap (in rate) to capacity. The new bounded complexity result is achieved by puncturing bits, and allowing in this way a sufficient number of state nodes in the Tanner graph representing the codes. We also derive an information-theoretic lower bound on the decoding complexity of randomly punctured codes on graphs. The bound holds for every memoryless binary-input output-symmetric channel, and is refined for the BEC.