De-randomizing Shannon: The Design and Analysis of a Capacity-Achieving Rateless Code

TL;DR

This paper introduces spinal codes, a new class of rateless codes that achieve Shannon capacity for BSC and AWGN channels using a sequential hash-based structure, with proven efficiency and capacity-achieving properties.

Contribution

The paper proves that spinal codes are capacity-achieving rateless codes for BSC and AWGN channels with efficient encoding and decoding, using a novel hash-based sequential structure.

Findings

Achieve Shannon capacity for BSC and AWGN channels.

First rateless codes with proven capacity for these channels.

Efficient polynomial-time encoder and decoder.

Abstract

This paper presents an analysis of spinal codes, a class of rateless codes proposed recently. We prove that spinal codes achieve Shannon capacity for the binary symmetric channel (BSC) and the additive white Gaussian noise (AWGN) channel with an efficient polynomial-time encoder and decoder. They are the first rateless codes with proofs of these properties for BSC and AWGN. The key idea in the spinal code is the sequential application of a hash function over the message bits. The sequential structure of the code turns out to be crucial for efficient decoding. Moreover, counter to the wisdom of having an expander structure in good codes, we show that the spinal code, despite its sequential structure, achieves capacity. The pseudo-randomness provided by a hash function suffices for this purpose. Our proof introduces a variant of Gallager's result characterizing the error exponent of random codes for any memoryless channel. We present a novel application of these error-exponent results within the framework of an efficient sequential code. The application of a hash function over the message bits provides a methodical and effective way to de-randomize Shannon's random codebook construction.