From Bit to Block: Capacity Achievement via Product Coding

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.