4488 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 10, OCTOBER 2008 List Decoding of Biorthogonal Codes and the Hadamard Transform With Linear Complexity Ilya Dumer, Fellow, IEEE, Grigory Kabatiansky, and Cédric Tavernier Abstract—Let a biorthogonal Reed–Muller code of length be used on a memoryless channel with an input al- phabet and a real-valued output . Given any nonzero received vector in the Euclidean space and some parameter , our goal is to perform list decoding of the code and re- trieve all codewords located within the angle from . For an arbitrarily small , we design an algorithm that outputs this list of codewords with the linear complexity order of bit oper- ations. Without loss of generality, let vector be also scaled to the Euclidean length of the transmitted vectors. Then an equiva- lent task is to retrieve all coefficients of the Hadamard transform of vector whose absolute values exceed . Thus, this decoding algorithm retrieves all -significant coefficients of the Hadamard transform with the linear complexity instead of the com- plexity of the full Hadamard transform. Index Terms—Biorthogonal codes, Hadamard transform, soft- decision list decoding. I. INTRODUCTION B IORTHOGONAL (first-order) Reed–Muller codes have been extensively used in communica- tions and addressed in many papers since the 1960s. These codes have optimal parameters and achieve the maximum possible distance for the given length and dimension . One renowned decoding algorithm designed by Green [1] performs maximum-likelihood decoding of codes and finds the distances from the received vector to all codewords of with complexity of bit operations. Another algorithm designed by Litsyn and Shekhovtsov [2] performs bounded distance decoding and corrects up to errors with linear complexity . In the area of probabilistic decoding, a major breakthrough has been achieved by Goldreich and Levin [3]. Their algorithm takes any received vector and outputs the list of codewords of within a decoding radius performing this task with a high probability and a low poly-logarithmic complexity for any and . Recently, Manuscript received July 2, 2007. Current version published September 17, 2008. The work of I. Dumer was supported in part by the National Science Foundation under Grants CCF0622242 and CCF0635339. The work of G. Ka- batiansky was supported in part by the Russian Foundation for Fundamental Re- search under Grants 06-01-00226 and 08-07-92495. The material in this paper was presented in part at the IEEE International Symposium on Information Theory, Nice, France, June 2007. I. Dumer is with the Department of Electrical Engineering, University of Cal- ifornia, Riverside, CA 92521 USA (e-mail: dumer@ee.ucr.edu). G. Kabatiansky is with the Institute for Information Transmission Problems, Moscow 101447, Russia and with INRIA, Rocquencourt, France (e-mail: kaba@iitp.ru). C. Tavernier is with Communications and Systems (CS), Le Plessis Robinson, France; (e-mail: tavernier.cedric@gmail.com). Communicated by T. Etzion, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2008.929014 list decoding of codes has been extended to deter- ministic algorithms. In particular, the algorithm of [4] performs error-free list decoding within the radius with linear complexity for any received vector. This paper advances the results of [4] in two different di- rections. First, we extend list decoding of codes to an arbitrary memoryless semi-continuous channel. Second, the former complexity of [4] will be reduced to . In doing so, we use the following setup. Let a binary vector be mapped onto the Euclidean vector with symbols . Given two binary vectors and , consider the Hamming dis- tance , the Euclidean distance , and the inner product of their maps . Then (1) Now any binary code is mapped into the cube , which in turn belongs to the Euclidean sphere of radius in the Euclidean space . Thus, any binary code of Hamming distance becomes a spherical code, where two different codewords have the inner product at most and the angle at least . Below, we consider a memoryless channel with an input al- phabet and some larger output alphabet (usually, ). We use a code on this channel and replace an output in any position with its log-likelihood ratio We then call a received vector. Note that any codeword has a higher posterior probability than another codeword if it also has a larger inner product . Note that all codewords become equiprobable for therefore, we will assume that . Without loss of generality, we can multiply by the scalar , where is the squared Euclidean length of vector Then, all vectors and belong to the same sphere We now proceed with the biorthogonal codes. Let be any affine Boolean function defined on all points 0018-9448/$25.00 © 2008 IEEE Authorized licensed use limited to: Univ of Calif Riverside. Downloaded on March 03,2010 at 21:09:31 EST from IEEE Xplore. Restrictions apply.