Punctured Low-Bias Codes Behave Like Random Linear Codes
Venkatesan Guruswami, Jonathan Mosheiff
TL;DR
The work demonstrates that puncturing a short, low-bias mother code yields a code whose local, combinatorial properties closely match those of random linear codes, effectively derandomizing RLCs with only linear randomness in the length. By framing properties as monotone, local, and row-symmetric, the authors transfer the RLC threshold behavior to punctured codes and obtain capacity- and list-decodability results that apply to both worst-case and stochastic error models. The results extend to near-optimal-distance mother codes and large alphabets, and include a general derandomization technique for sampling RLC-like codes with guaranteed property preservation. This derandomization has practical implications for constructing capacity-achieving codes with algebraic structure and potentially enabling efficient decoding in structured settings.
Abstract
Random linear codes are a workhorse in coding theory, and are used to show the existence of codes with the best known or even near-optimal trade-offs in many noise models. However, they have little structure besides linearity, and are not amenable to tractable error-correction algorithms. In this work, we prove a general derandomization result applicable to random linear codes. Namely, in settings where the coding-theoretic property of interest is "local" (in the sense of forbidding certain bad configurations involving few vectors -- code distance and list-decodability being notable examples), one can replace random linear codes (RLCs) with a significantly derandomized variant with essentially no loss in parameters. Specifically, instead of randomly sampling coordinates of the (long) Hadamard code (which is an equivalent way to describe RLCs), one can randomly sample coordinates of any code with low bias. Over large alphabets, the low bias requirement can be weakened to just large distance. Furthermore, large distance suffices even with a small alphabet in order to match the current best known bounds for RLC list-decodability. In particular, by virtue of our result, all current (and future) achievability bounds for list-decodability of random linear codes extend automatically to random puncturings of any low-bias (or large alphabet) "mother" code. We also show that our punctured codes emulate the behavior of RLCs on stochastic channels, thus giving a derandomization of RLCs in the context of achieving Shannon capacity as well. Thus, we have a randomness-efficient way to sample codes achieving capacity in both worst-case and stochastic settings that can further inherit algebraic or other algorithmically useful structural properties of the mother code.