Soft-decision list decoding of Reed-Muller codes with linear complexity Ilya Dumer 1 University of California at Riverside Riverside, CA, USA Email: dumer@ee.ucr.edu Grigory Kabatiansky 2 Inst. for Info. Transmission Problems Moscow, Russia Email: kaba@iitp.ru C´ edric Tavernier National Knowledge Center (NKC-EAI) Abu-Dhabi, UAE Email: tavernier.cedric@gmail.com Abstract—Let a binary Reed-Muller code RM(s, m) of length n be used on a memoryless channel with an input alphabet ±1 and a real-valued output R. Given a received vector y in R n , we define its generalized distance T to any codeword c as the sum ∑ |yj | taken over all positions j, in which vectors y,c have opposite signs. We then consider the list LT of codewords located within distance T from the received vector y and estimate the size LT of this list using the generalized Johnson bound. For any RM code RM(s, m) of fixed order s, the algorithm is proposed that performs list decoding beyond the error-correcting radius with linear complexity in length n and retrieves the code list LT with complexity of order nsLT for any decoding radius T within the generalized Johnson bound. I. I NTRODUCTION Preliminaries. Reed-Muller (RM) codes RM (s, m) are de- fined by any nonnegative integers m, s, where m ≥ s, and have length n, dimension k, and distance d as follows: n =2 m , k = s X i=0 ( m i ) , d =2 m-s . RM codes have been extensively studied since 1950s thanks to their simple code structure and fast decoding procedures. In particular, majority decoding proposed in the seminal paper [5] has complexity order at most nk. Even a lower complexity order of n min(s, m - s) is required for recursive techniques [8], [9]. These algorithms also correct many errors beyond the error-correcting radius of d/2. In particular, for long RM codes RM (s, m) of any given order s and for an arbitrarily small parameter ε> 0, majority decoding [6] and the recursive technique [9] correct most error patterns of weight up to T = n(1 - ε)/2. However, these algorithms depend on a specific error pattern of weight T . As a result, these algorithms cannot necessarily output the complete list of codewords located within any distance T ≥ d/2 from a given vector y. By contrast, the recent papers [2], [3], [4] yield the complete list of codewords within a specified radius T <d for a long code RM (s, m) of a given order s but require a higher complexity of order O(n ln s-1 n) or above. In this paper, we reduce this complexity order of n ln s-1 n to the linear order O(n) for any radius T bounded by the Johnson bound. We 1 Research was supported by NSF grant ECCS-11043129 and ARO grant W911NF-11-1-0027 2 Research was supported by RFFI grants 09-01-00536 and 11-01-00735 of the Russian Foundation for Fundamental Research will also extend these results to an arbitrary memoryless semi- continuous channel. We assume that the codewords (c 0 , ..., c n-1 ) are taken from {±1} n and a received vector z belongs to the Euclidean space R n . Then we consider the vector y =(y 0 , ..y n-1 ) of log likelihoods y j = ln Pr{c j = +1|z j } Pr{c j = -1|z j } Here we define an error cost ω j = |y j | in each position j . In the sequel, all sums over integers j will be taken in the entire range [0,n - 1] if not stated otherwise. Without loss of generality, we scale vector y to have the same squared length ∑ j y 2 j = n as that of any binary vector in {±1} n . We also say that y has generalized weight W = ∑ j ω j . Given y ∈ R n and any vector c ∈ {±1} n , consider the set J (y,c)= {j : y j c j < 0} of positions, where vectors y,c have opposite signs. Then we introduce the generalized Hamming distance D(y,c)= X J (y,c) ω j . (1) Equivalently, D(y,c) = W/2 -hy,ci/2, where ha, bi = ∑ j a j b j denotes the inner product of any two real valued vec- tors a, b. Clearly, the input c with the smallest distance D(y,c) yields the vector with the maximum posterior probability. In the sequel, we estimate decoding complexity by the number of operations with real numbers. Our main result is defined for semicontinuous channels with the generalized Hamming distance D(y,c) and reads as follows. Theorem 1: Consider Reed-Muller code RM(s, m) of a fixed order s. Let y ∈ R n be a received vector of generalized weight W and let T max = W/2 - p (n - 2d)n/2. (2) Then for any generalized distance T<T max , RM code C can be decoded into the code list L T ;C (y)= {c ∈ C : D(y,c) ≤ T } (3) with complexity at most 2nL T min(s, m - s), (4) where L T = 2dn (W - 2T ) 2 - (n - 2d)n . (5) 2011 IEEE International Symposium on Information Theory Proceedings 978-1-4577-0594-6/11/$26.00 ©2011 IEEE 2214