Improved List-Decodability of Random Linear Binary Codes

Abstract

There has been a great deal of work establishing that random linear codes are as list-decodable as uniformly random codes, in the sense that a random linear binary code of rate 1 − H(p) − is (p, O(1/))-list-decodable with high probability. In this work, we show that such codes are (p, H(p)/ + 2)-list-decodable with high probability, for any p ∈ (0, 1/2) and > 0. In addition to improving the constant in known list-size bounds, our argument-which is quite simple-works simultaneously for all values of p, while previous works obtaining L = O(1/) patched together different arguments to cover different parameter regimes. Our approach is to strengthen an existential argument of (Guruswami, Håstad, Sudan and Zuckerman, IEEE Trans. IT, 2002) to hold with high probability. To complement our upper bound for random linear binary codes, we also improve an argument of (Guruswami, Narayanan, IEEE Trans. IT, 2014) to obtain a tight lower bound of 1/ on the list size of uniformly random binary codes; this implies that random linear binary codes are in fact more list-decodable than uniformly random binary codes, in the sense that the list sizes are strictly smaller. To demonstrate the applicability of these techniques, we use them to (a) obtain more information about the distribution of list sizes of random linear binary codes and (b) to prove a similar result for random linear rank-metric codes.

List Decoding Random Euclidean Codes and Infinite Constellations

“…The conjecture is known to be true over F 2 [10]. However, it is open in other cases if we insist the universal constant in the list size to be one.…”

“…Then by the continuity of the entropy function, the capacity is very low and can be a vanishing function of δ as δ Ñ 0. In this case, many existing list decodability results for 1 Actually, Li and Wootters [10] showed that for any constant γ ą 0, the list size of a random code is bounded from below by 1´γ δ whp. 2 Rigorously, there are two commonly used definitions of explicitness in the literature.…”

“…Indeed, it is wellknown and easy to show that the list size of a random code (of which each codeword is sampled uniformly and independently) of rate 1 ´Hq ppq ´δ against a power-p adversary is at most 1{δ with high probability (whp). It also turns out [9], [10] that 1{δ is the correct scaling for the list size of a random code. Namely, there is an essentially matching 1 lower bound 1{δ via second moment method.…”

“…Noise level Rate List size whp / with constant probability / existential Reference q ě 2 p P p0, 1 ´1{qq R " 1 ´Hqppq ´δ L " q Op1{δq whp Folklore q " 2 p P p0, 1{2q R " 1 ´Hppq ´δ L ď 1{δ existential [12] q ě 2 p P p0, 1 ´1{qq R " 1 ´Hqppq ´δ L ď Cp,q{δ whp [11] q ě 2 p " p1 ´1{qqp1 ´τ q R " Ω ´τ2 log 3 pq{τ q log q ¯L " Op1{τ 2 q with constant probability [13] q ě 2 p " p1 ´1{qqp1 ´τ q R " Ωpτ 2 { log qq L " Op1{τ 2 q whp [14] q " 2 p P p0, 1{2q R " 1 ´Hppq ´δ L ď Hppq{δ `2 whp [10] TABLE I: A non-exhaustive list of the state of the art of list decodability of random linear codes.…”

“…This is an improvement on [13] via similar techniques and also fills in the gap in [11] for large p. 6) Rudra and Wootters [15], [16], [17] employed heavy machinery of generic chaining to provide improved bounds when the field size is very large, say, allowed to scale with 1{δ and even n. The parameter regimes become complicated in this situation and we do not copy them here. 7) Li and Wootters [10] showed that a random binary linear code of rate 1 ´Hppq ´δ is pp, Hppq{δ `2q-list decodable whp for any p P p0, 1{2q and δ ą 0. They did so by lifting the existential result in [12] to a whp one.…”