List decoding of Reed-Muller codes up to the Johnson bound with almost linear complexity Ilya Dumer

Grigory Kabatiansky

C´edric Tavernier1

University of California Riverside, CA, USA Email: [email protected]

Inst. for Information Transmission Problems Moscow, Russia Email: [email protected]

THALES Communications Colombes, France Email: [email protected]

Abstract— A new deterministic list decoding algorithm is proposed for general Reed-Muller codes RM (s, m) of length n = 2m and distance d = 2m−s . Given n and d, the algorithm performs beyond the bounded distance threshold of d/2 and has a low complexity order of nms−1 for any decoding radius T that is less than the Johnson bound.

I. I NTRODUCTION Preliminaries. Reed-Muller (RM) codes RM (s, m) - defined by two integers m and s, where m ≥ s ≥ 0 - have length n, dimension k, and distance d as follows: n = 2m ,

k=

s X

(m i ),

d = 2m−s .

i=0

Over the past 50 years, there have been many studies of RM codes thanks to their simple code structure and fast decoding procedures. In particular, majority decoding proposed in the seminal paper [1] has complexity order of nk or less for any code RM (s, m). Even a lower complexity order of n min(s, m−s) is required for different recursive techniques of [12], [13], and [14]. These algorithms also correct many errors beyond the bounded-distance decoding (BDD) radius of d/2. In particular, for long RM codes RM (s, m) of any given order s, the majority decoding (see [2]) and the recursive technique (see [14]) correct most error patterns of weight T = n(1−ε)/2 or less for any ε > 0. However, for any given error weight T ≥ d/2, decoding performance depends on a specific error pattern. As a result, these algorithms cannot necessarily output the complete list of codewords within the radius T ≥ d/2. Another line of research focuses on list decoding algorithms and has originated in the paper [11]. Given code C and decoding radius T, a list decoder gets any received vector y and produces the list LT ;C (y) = {c ∈ C : d(y, c) ≤ T } of all code vectors c ∈ C located within the distance T from y. A particularly important question is to define the maximum decoding radius T that allows for low-complexity decoding of a specific code. As a first cut to this problem, we consider list decoding within the Johnson bound, which guarantees that the 1 The work of C. Tavernier was partially supported by DISCREET, IST project no. 027679, funded in part by the European Commission’s Information Society Technology 6th Framework Programme

output list LT ;C (y) will stay limited for growing blocklengths. This bound is defined as follows. Consider a code C with the minimum code distance at least d = δn. Let Bn,T (y) = {x : d(y, x) ≤ T } be the Hamming ball of radius T = ωn centered at any y. The Johnson bound limits the number of codewords δ |LT ;C (y)| = |C ∩ Bn,T (y)| ≤ (1) δ − 2ω(1 − ω) provided that (1) has positive denominator. In particular, this bound shows that any code RM (s, m) contains at most a constant number δ |LT ;RM (s,m) (y)| ≤ (2) 2²(1 − 2Js + ²) of codewords located within a decoding radius T² (s) = n(Js − ²) from any center y, where p Js = 2−1 (1 − 1 − 2−s+1 ),

(3)

² ∈ (0, Js ].

Note that for any s, 2−(s+1) + 2−(2s+2) ≤ Js ≤ 2−(s+1) + 2−(2s+1) . Therefore for any code RM (s, m), nJs is larger than the BDD threshold of d/2. Summary of the results. In this presentation, our results are summarized in the following theorem. Theorem 1: For any received vector y and any ² > 0, list decoding of codes RM (s, m) can be performed within the decoding radius n(Js − ²) with complexity ( λ1 n²−3 , if s = 1, χ(m, s, ²) ≤ (4) −2 s−1 λs ² mn + µs m n, if s ≥ 2. where the constants λs and µs depend on the order s only. Let us compare Theorem 1 with other results known for list decoding or RM codes. The first significant breakthrough in this area was achieved in the paper [8] that presents a very efficient - though probabilistic - algorithm for codes RM (1, m) of the first order. Later, the result was substantially extended in [9] and [10] to a larger context of nonbinary codes. Note that the algorithms of [8] - [10] are not error-specific, and output the required lists with high fidelity regardless of

its input y. On the other hand, these algorithms output the required list of codewords with some small error probability Perr = P1 + P2 . Here P1 is the probability that a “good” code vector c (i.e. d(y, c) ≤ T ) is discarded from the output list, and P2 is another probability of leaving a “bad” code vector within this list. For binary codes RM (1, m), the algorithms of [8] - [10] have decoding radius T² (1) = n(1/2 − ²) and −1 complexity poly(log n, ²−1 , logPerr ). To date, the question of constructing (probabilistic) list decoding algorithm with polylog complexity is open for all binary RM codes of order s > 1. Theorem 1 can be extended to probabilistic decoding that performs within the radius T² (s) with polylog complexity for any given s; however, the proof is beyond the scope of this correspondence. Our conjecture is that such a decoding with polylog complexity does not exist if the decoding radius is larger than the Johnson bound. Another - deterministic - list decoding algorithm is proposed in [5] for general RM codes RM (s, m). This is done by applying the Guruswami-Sudan (GS) algorithm [4] to the 1shortened codes RM 0 (s, m) of length 2m − 1. This decoding design is due to the two facts. First, 1-shortened codes RM 0 (s, m) are subcodes of BCH-codes of the same designed distance d. Second, BCH codes are the subfield subcodes of RS codes, again of the same distance d. These two facts lead to a list decoding algorithm [5] that has decoding radius

is a Boolean function of degree at most s and (x1 , ..., xm ) range over all 2m points of the Boolean cube of dimension m. Let y be the received vector. Our goal is to design the list L² (y) = {f ∈ RM (s, m) : d(y, f ) ≤ T = n(Js − ²)}

(5)

whose size is restricted in (2). First, we wish to recover all terms of degree s in f (x) ∈ L² (y). For any list f ∈ L² (y), consider the corresponding list of (different) polynomials def

=

X

L²,high (y) = {f(high) fi1 ,...,is xi1 ...xis |f ∈ L² (y)}.

1≤i1