Random Access Expectation in DNA Storage and Fountain Codes

TL;DR

This paper analyzes the expected number of coded symbols needed for decoding in DNA storage using symmetric linear codes, revealing near-optimal bounds related to LT codes.

Contribution

It establishes an equivalence between binary fully symmetric codes and LT codes, and analyzes their decoding expectations under a peeling decoder in large blocklengths.

Findings

Expected number of symbols needed is at least π/4 (~0.7854)

Achievable expectation is approximately 0.7869

Analysis applies to large blocklength limits

Abstract

Motivated by DNA data storage, we study the expected number of coded symbols drawn from a linear code until a desired information symbol can be decoded - the random access expectation. We focus on generator matrices with a type of symmetry, conjectured in prior work to be optimal, which we call fully symmetric. We point out an equivalence between binary fully symmetric codes and LT codes. Using this observation, we analyze the random access expectation of binary fully symmetric codes under a peeling decoder, in the large blocklength limit. Under these assumptions, the random access expectation, normalized by the number of information symbols, is at least $\pi/4 \approx 0.7854$, while a value of $\approx 0.7869$ is achievable.