Random Shortening of Linear Codes and Applications

TL;DR

This paper studies the process of random shortening of linear codes, showing it produces biased codes with good properties from weaker initial codes, complementing previous work on puncturing.

Contribution

It introduces and analyzes the dual operation of random shortening, demonstrating its effectiveness in producing high-quality codes from weaker initial codes.

Findings

Random shortening yields ε-biased codes with high probability.

The method works over any field size and maintains constant rate.

Proofs involve novel weight distribution estimation techniques.

Abstract

Random linear codes (RLCs) are well known to have nice combinatorial properties and near-optimal parameters in many different settings. However, getting explicit constructions matching the parameters of RLCs is challenging, and RLCs are hard to decode efficiently. This motivated several previous works to study the problem of partially derandomizing RLCs, by applying certain operations to an explicit mother code. Among them, one of the most well studied operations is random puncturing, where a series of works culminated in the work of Guruswami and Mosheiff (FOCS' 22), which showed that a random puncturing of a low-biased code is likely to possess almost all interesting local properties of RLCs. In this work, we provide an in-depth study of another, dual operation of random puncturing, known as random shortening, which can be viewed equivalently as random puncturing on the dual code. Our main results show that for any small $\varepsilon$, by starting from a mother code with certain weaker conditions (e.g., having a large distance) and performing a random (or even pseudorandom) shortening, the new code is $\varepsilon$-biased with high probability. Our results hold for any field size and yield a shortened code with constant rate. This can be viewed as a complement to random puncturing, and together, we can obtain codes with properties like RLCs from weaker initial conditions. Our proofs involve several non-trivial methods of estimating the weight distribution of codewords, which may be of independent interest.