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.

翻译：随机线性码（RLCs）因其优异的组合性质和在多种场景下接近最优的参数而广为人知。然而，构造与RLCs参数匹配的显式编码极具挑战性，且RLCs难以高效译码。这促使以往多项研究通过将特定操作应用于显式母码来部分去随机化RLCs。其中研究最深入的操作之一是随机删余：一系列工作最终由Guruswami和Mosheiff（FOCS'22）完成，证明低偏置码的随机删余很可能保留RLCs几乎所有有趣的局部性质。本文深入研究了随机删余的对偶操作——随机缩短，该操作可等价视为对偶码的随机删余。主要结果表明：对于任意小量$\varepsilon$，从满足特定较弱条件（如具有大距离）的母码出发，执行随机（甚至伪随机）缩短后，新码能以高概率实现$\varepsilon$-偏置。该结果适用于任意域大小，且生成的缩短码具有恒定码率。本研究可视为随机删余的补充，两者结合可从更弱的初始条件获得具有RLCs性质的码。证明过程中涉及若干估计码字重量分布的非平凡方法，这些方法可能具有独立研究价值。