Reed-Muller Codes Achieve Capacity on Erasure Channels (1505.05123v2)

Abstract: This paper introduces a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise structure of the codes, this method requires only that the codes are highly symmetric. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge to a number between 0 and 1, and the permutation group of each code is doubly transitive. This also provides a rare example in information theory where symmetry alone implies near-optimal performance. An important consequence of this result is that a sequence of Reed-Muller codes with increasing blocklength achieves capacity if its code rate converges to a number between 0 and 1. This possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1. Moreover, these results extend naturally to affine-invariant codes and, thus, to all extended primitive narrow-sense BCH codes. The primary tools used in the proof are the sharp threshold property for monotone boolean functions and the area theorem for extrinsic information transfer functions.

• The paper proves that Reed-Muller codes achieve capacity on erasure channels using maximum a posteriori decoding by leveraging the inherent symmetry of the codes.
• The authors employ methods based on doubly transitive permutation groups and EXIT function analysis to demonstrate that code symmetry ensures capacity-achieving performance.
• A key finding is that code symmetry alone can suffice for achieving capacity on erasure channels, extending this property to other affine-invariant codes.

Insights into Reed-Muller Codes Achieving Capacity on Erasure Channels

The paper presented by Santhosh Kumar and Henry D. Pfister introduces a novel approach proving that certain sequences of deterministic linear codes, specifically Reed-Muller codes, achieve capacity on erasure channels when decoded using maximum a posteriori (MAP) methods. Significant attention is given to the inherent symmetry of these codes, which allows them to reach near-optimal performance without reliance on intricate structural properties.

Key Contributions

The authors employ a strategy that leverages the symmetry properties of linear codes, particularly focusing on sequences where the permutation group of the codes is doubly transitive. Doubly transitive groups—a stringent form of symmetry—ensure that the codes are invariant under any permutation of code bits that preserves the distances between them. This property is crucial since it implies that the performance of such codes can be analyzed and predicted accurately without the need for detailed structural insights.

A central claim of the paper is that Reed-Muller codes, when considered as a sequence with increasing blocklengths and rates converging to a number between 0 and 1, achieve capacity on erasure channels. This is in contrast to prior results, which only affirmed this property under limiting rates of 0 or 1. The paper extends these results to other affine-invariant codes, such as extended primitive narrow-sense BCH codes.

Theoretical Foundation

The primary toolset employed by the authors includes the theory of EXIT (extrinsic information transfer) functions and the concept of sharp threshold properties in monotone Boolean functions. They detail how the EXIT functions, reflecting the conditional entropy of input bits given other bits' output, transition in a highly controlled manner due to the code symmetries. The EXIT area theorem plays a critical role here, determining the function's area under the curve and thus conferring the threshold characteristics required for capacity-achieving performance.

Numerical Results and Strong Claims

• Symmetry Alone Suffices: The paper provides a rare insight where symmetry, rather than detailed structural code design, is the key factor leading to capacity-achieving behavior.
• Sharp Thresholds: The results show sharp threshold behavior for the EXIT functions, meaning the codes switch abruptly from non-capacity achieving to capacity achieving as the channel conditions vary, a pivotal property for ensuring reliable communication near the theoretical capacity limit.
• Extension to Other Codes: By proving that all affine-invariant codes share this capacity-achieving property, the authors broaden the applicability of their method far beyond Reed-Muller codes alone.

Implications and Future Directions

The constructs and proofs extend primarily to erasure channels; however, by conceptual extrapolations, they could potentially be adapted to broader coding scenarios, such as binary symmetric channels. The rigorous approach taken in their proofs sets a solid foundation for exploring these extensions, pushing the boundaries of current theory.

This work reignites interest in traditional codes like Reed-Muller, which have been somewhat eclipsed by modern codes such as turbo and LDPC but remain a focal point in areas like cryptography and locality-sensitive applications. Moreover, the approach may inspire future research into other code structures with high inherent symmetries, posing questions about their potential undiscovered properties.

Ultimately, this paper reinforces the notion that symmetry in code design is not merely a triviality but can be central to achieving performance goals, offering a pathway to further theoretical exploration in coding theory where conventional wisdom falls short.