From Bit to Block: Capacity Achievement via Product Coding

2026-07-07

AI Summary

This work addresses the challenge of elevating bit-level capacity-achieving coding schemes to block-level reliability. It proposes a product code construction that combines low-error-probability row codes with high-rate column codes: the channel output is first cleaned via row decoding, and the remaining sparse errors are then corrected using column codes. Through rigorous theoretical analysis employing bounded-distance decoding, binomial large-deviation estimates, and union bound techniques, the paper establishes—for the first time—that block error probability can be driven to zero while achieving the capacity of any fixed binary memoryless symmetric (BMS) channel. The proposed family of RM–BCH product codes provides a concrete and practical instantiation of this approach.

Abstract

This paper shows that product coding can convert bit-level reliability into block-level reliability at the same asymptotic rate. The encoding process combines a row code of the target rate with vanishing bit-error probability and a high-rate column code with bounded-distance correction capability. The decoding process first decodes rows, thereby purifying the channel output into a sparse residual error pattern, and then decodes columns to clean the residual errors. We prove that, if the column correction radius exceeds the residual bit-error probability and the column length is large enough for a binomial large-deviation bound to overcome the union bound over columns, then the product code has vanishing block-error probability. Thus a bit-level capacity-achieving family can be converted into a block-level capacity-achieving product-code family. As an application, we construct an RM--BCH product-code family that achieves the capacity of any fixed BMS channel.

Problem

• product coding
• bit-level reliability
• block-level reliability
• capacity achievement
• block-error probability

Innovation

• product coding
• capacity achievement
• block-error probability
• BMS channel
• large-deviation bound