List-decoding reed-muller codes over small fields

Abstract

We present the first local list-decoding algorithm for the r th order Reed-Muller code RM(r, m) over F2 for r ≥ 2. Given an oracle for a received word R : F m 2 → F2, our randomized local list-decoding algorithm produces a list containing all degree r polynomials within relative distance (2 −r − ε) from R for any ε > 0 in time poly(m r , ε −r ). The list size could be exponential in m at radius 2 −r , so our bound is optimal in the local setting. Since RM(r, m) has relative distance 2 −r , our algorithm beats the Johnson bound for r ≥ 2. In the setting where we are allowed running-time polynomial in the block-length, we show that list-decoding is possible up to even larger radii, beyond the minimum distance. We give a deterministic list-decoder that works at error rate below J(2 1−r ), where J(δ) denotes the Johnson radius for minimum distance δ. This shows that RM(2, m) codes are list-decodable up to radius η for any constant η < 1 2 in time polynomial in the block-length.Over small fields Fq, we present list-decoding algorithms in both the global and local settings that work up to the list-decoding radius. We conjecture that the list-decoding radius approaches the minimum distance (like over F2), and prove this holds true when the degree is divisible by q − 1.

List Decoding Tensor Products and Interleaved Codes

“…This result is interesting in light of a conjecture by Gopalan et al stating that Reed-Muller codes of degree m over F q are list-decodable up to the minimum distance (they proved this result for q = 2) [12]. Our result shows m-wise Hadamard tensors which are a natural sub code of order m Reed-Muller codes (with better distance but lower rate) are indeed list-decodable up to the minimum distance.…”

“…Therefore, for r = r(δ, η) large enough, the number of codewords in a Hamming ball of radius η < δ whose pairwise differences all have rank > r can be bounded from above using the Johnson bound. Using the deletion argument from [12], the task of bounding the list-size for radius η now reduces to bounding the number of rank r codewords within radius η. We accomplish this task, for both interleaved and product codes, using additional combinatorial ideas.…”

“…This study has had substantial impact on other areas such as complexity theory [30,32], cryptography [11,1] and computational learning [19,21]. Examples of codes which are known to be list-decodable beyond the Johnson bound have been rare: Extractor codes [31,14], folded Reed-Solomon codes [24,17], group homomorphisms [6] and Reed-Muller codes over small fields [12] are the few examples known to us.…”

Efficiently decoding Reed-Muller codes from random errors

“…This question received a lot of attention and among the works in this area we mention the seminal works of Goldreich and Levin on Hadamard Codes [GL89] and of Sudan [Sud97] and Guruswami and Sudan [GS99] on list decoding Reed-Solomon codes. Recently, the list-decoding question for Reed-Muller codes was studied by Gopalan, Klivans and Zuckerman [GKZ08] and by Bhowmick and Lovett [BL15], who proved that the list decoding radius 1 of Reed-Muller codes, over F 2 , is at least twice the minimum distance (recall that the unique decoding radius is half that quantity) and is smaller than four times the minimal distance, when the degree of the code is constant.…”

An HARQ Based Optimized Error Correction Technique

“…BCH code [3,4,5],a multiple bit ECC, is a further generalization of hamming code where 2m-1 bits comprising of mt redundant bits enable correction up to t-bits error. Other significant ECC include Reed Muller code [6,7], Low density parity check code (LDPC) [8,9,10] and turbo code [11,12]. FEC has reliability issues and ARQ suffers with degrading throughput.…”