Capacity-achieving ensembles for the binary erasure channel with bounded complexity

TL;DR

This paper introduces new sequences of irregular repeat-accumulate codes that achieve capacity on the binary erasure channel with bounded complexity per bit, unlike previous codes whose complexity grows with the inverse gap to capacity.

Contribution

The paper presents novel capacity-achieving code ensembles with bounded complexity, achieved through puncturing and Tanner graph modifications, and derives a lower bound on decoding complexity.

Findings

Achieves capacity with bounded complexity per information bit.

Uses puncturing to control Tanner graph complexity.

Provides a 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.