Deterministic Rateless Codes for BSC

TL;DR

This paper introduces the first deterministic rateless code for the binary symmetric channel, achieving capacity with efficient encoding, decoding, and extremely low error probability, improving upon previous probabilistic codes.

Contribution

The paper constructs a deterministic rateless code for BSC with efficient encoding/decoding and near-exponential error decay, unlike prior probabilistic approaches.

Findings

Constructed a deterministic rateless code for BSC.

Achieves near-capacity communication with low error probability.

Encoding and decoding run in near-logarithmic time.

Abstract

A rateless code encodes a finite length information word into an infinitely long codeword such that longer prefixes of the codeword can tolerate a larger fraction of errors. A rateless code achieves capacity for a family of channels if, for every channel in the family, reliable communication is obtained by a prefix of the code whose rate is arbitrarily close to the channel's capacity. As a result, a universal encoder can communicate over all channels in the family while simultaneously achieving optimal communication overhead. In this paper, we construct the first \emph{deterministic} rateless code for the binary symmetric channel. Our code can be encoded and decoded in $O(\beta)$ time per bit and in almost logarithmic parallel time of $O(\beta \log n)$, where $\beta$ is any (arbitrarily slow) super-constant function. Furthermore, the error probability of our code is almost exponentially small $\exp(-\Omega(n/\beta))$. Previous rateless codes are probabilistic (i.e., based on code ensembles), require polynomial time per bit for decoding, and have inferior asymptotic error probabilities. Our main technical contribution is a constructive proof for the existence of an infinite generating matrix that each of its prefixes induce a weight distribution that approximates the expected weight distribution of a random linear code.