On codes decoding a constant fraction of errors on the BSC

Abstract

Using techniques and results from [8] we strengthen the bounds of [10] on the weight distribution of linear codes achieving capacity on the BEC. In particular, we show that for any doubly transitive binary linear code C ⊆ {0, 1} n of rate 0 < R < 1 with weight distribution (a 0 , ..., a n ) holdsFor doubly transitive codes with minimal distance at least Ω (n c ), 0 < c ≤ 1, the error factor of 2 o(n) in this bound can be removed at the cost of replacing 1 − R with a smaller constant a = a(R, c) < 1 − R. Moreover, in the special case of Reed-Muller codes, due to the additional symmetries of these codes, this error factor can be removed at essentially no cost.This implies that for any doubly transitive code C of rate R with minimal distance at least Ω (n c ), there exists a positive constant p = p(R, c) such that C decodes errors on BSC(p) with high probability if p < p(R, c). For doubly transitive codes of a sufficiently low rate (smaller than some absolute constant) the requirement on the minimal distance can be omitted, and hence this critical probability p(R) depends only on R. Furthermore, p(R) → 1 2 as R → 0. In particular, a Reed-Muller code C of rate R decodes errors on BSC(p) with high probability if1 4 ln 2 , answering a question posed in [1].

An Upper Bound on $\ell_q$ Norms of Noisy Functions

“…One way to quantify the decrease in the ℓ q norm of a function when this function is acted on by the noise operator was suggested in [4], where the ℓ q norm of the 'noisy version' of f was upperbounded by the average ℓ q norm of conditional expectations of f , given sets whose elements are chosen at random with certain explicit probability λ, depending on q and on ǫ. Some applications of this inequality were described in [4,5]. In this note we prove this inequality for integer q ≥ 2 with a slightly better (smaller) parameter λ, which leads to corresponding improvement in the applications.…”

“…It should be mentioned that the onedimensional claim turns out to be rather difficult, and we are only able to prove it for integer q ≥ 2. On the other hand, all the applications which we mention here (and in [4,5] as well) follow from the special case q = 2, which is much easier to prove (see Lemma 2.3). Theorem 1.1 makes it possible to improve the parameters in the results in [4,5] which use the inequality in [4].…”

“…On the other hand, all the applications which we mention here (and in [4,5] as well) follow from the special case q = 2, which is much easier to prove (see Lemma 2.3). Theorem 1.1 makes it possible to improve the parameters in the results in [4,5] which use the inequality in [4]. We state some of these results, with the new parameters.…”

“…We prove Theorem 1.1 in Section 2. Propositions 1.3-1.5 do not require new proofs since their claims are obtained by substituting the new value of λ from Theorem 1.1 in the corresponding claims in [4,5]. Note that Proposition 1.3 corresponds to Lemma 1.8 in [4], and Propositions 1.4-1.5 to Proposition 1.1 and Corollary 1.4 in [5].…”

“…The new inequality is tight for characteristic functions of subcubes. As an application, following [5], we show that a Reed-Muller code C of rate R decodes errors on BSC(p) with high probability ifThis is a (minor) improvement on the estimate in [5].…”