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Polar Codes are Optimal for Lossy Source Coding
"... We consider lossy source compression of a binary symmetric source with Hamming distortion function. We show that polar codes combined with a lowcomplexity successive cancellation encoding algorithm achieve the ratedistortion bound. The complexity of both the encoding and the decoding algorithm is ..."
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Cited by 63 (2 self)
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We consider lossy source compression of a binary symmetric source with Hamming distortion function. We show that polar codes combined with a lowcomplexity successive cancellation encoding algorithm achieve the ratedistortion bound. The complexity of both the encoding and the decoding algorithm is O(N log(N)), where N is the blocklength of the code. Our result mirrors Arıkan’s capacity achieving polar code construction for channel coding.
Grassmannian packings from operator ReedMuller codes
 IEEE Trans. Info. Theory
"... Abstract—This paper introduces multidimensional generalizations of binary Reed–Muller codes where the codewords are projection operators, and the corresponding subspaces are widely separated with respect to the chordal distance on Grassmannian space. Parameters of these Grassmannian packings are de ..."
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Abstract—This paper introduces multidimensional generalizations of binary Reed–Muller codes where the codewords are projection operators, and the corresponding subspaces are widely separated with respect to the chordal distance on Grassmannian space. Parameters of these Grassmannian packings are derived and a low complexity decoding algorithm is developed by modifying standard decoding algorithms for binary Reed–Muller codes. The subspaces are associated with projection operators determined by Pauli matrices appearing in the theory of quantum error correction and this connection with quantum stabilizer codes may be of independent interest. The Grassmannian packings constructed here find application in noncoherent wireless communication with multiple antennas, where separation with respect to the chordal distance on Grassmannian space guarantees closeness to the channel capacity. It is shown that the capacity of the noncoherent multipleinput–multipleoutput (MIMO) channel at both low and moderate signaltonoise ratio (SNR) (under the constraint that only isotropically distributed unitary matrices are used for information transmission) is closely approximated by these packings. Index Terms—Chordal distance, Grassmannian packings, noncoherent multipleinput–multipleoutput (MIMO) channel, Reed–Muller codes, spacetime codes. I.
SoftDecision Decoding of Reed–Muller Codes: A Simplified Algorithm
, 2006
"... oftdecision decoding is considered for general Reed–Muller (RM) codes of length and distance used over a memoryless channel. A recursive decoding algorithm is designed and its decoding threshold is derived for long RM codes. The algorithm has complexity of order � � and corrects most error patterns ..."
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Cited by 3 (0 self)
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oftdecision decoding is considered for general Reed–Muller (RM) codes of length and distance used over a memoryless channel. A recursive decoding algorithm is designed and its decoding threshold is derived for long RM codes. The algorithm has complexity of order � � and corrects most error patterns of the Euclidean weight of order � � instead of the decoding threshold P of the bounded distance decoding. Also, for long RM codes of fixed rate, the new algorithm increases 4/π times the decoding threshold of its harddecision counterpart.
List decoding of ReedMuller codes
 in &quot;Proceedings of ACCT’9
, 2004
"... We construct list decoding algorithms for first order ReedMuller codes RM[1, m] of length n = 2m correcting up to n ( 1 2 − ɛ) errors with complexity O(nɛ−3). Considering probabilistic approximation of these algorithms leads to randomized list decoding algorithms with characteristics similar to Gol ..."
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Cited by 2 (0 self)
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We construct list decoding algorithms for first order ReedMuller codes RM[1, m] of length n = 2m correcting up to n ( 1 2 − ɛ) errors with complexity O(nɛ−3). Considering probabilistic approximation of these algorithms leads to randomized list decoding algorithms with characteristics similar to GoldreichLevin algorithm, namely, of complexity O(m2ɛ−7 log 1
Simple maximumlikelihood decoding of generalized firstorder Reed–Muller codes
 IEEE Commun. Lett
, 2005
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SEE PROFILE
, 2016
"... Error exponents for two soft decision decoding algorithms of ReedMuller codes ..."
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Error exponents for two soft decision decoding algorithms of ReedMuller codes
Error Exponents for Recursive Decoding of Reed–Muller Codes on a BinarySymmetric Channel
"... Abstract—Error exponents are studied for recursive decoding of Reed–Muller (RM) codes and their subcodes used on a binarysymmetric channel. The decoding process is first decomposed into similar steps, with one new information bit derived in each step. Multiple recursive additions and multiplication ..."
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Abstract—Error exponents are studied for recursive decoding of Reed–Muller (RM) codes and their subcodes used on a binarysymmetric channel. The decoding process is first decomposed into similar steps, with one new information bit derived in each step. Multiple recursive additions and multiplications of the randomly corrupted channel outputs 6I are performed using a specific order of these two operations in each step. Recalculated random outputs are compared in terms of their exponential moments. As a result, tight analytical bounds are obtained for decoding error probability of the two recursive algorithms considered in the paper. For both algorithms, the derived error exponents almost coincide with simulation results. Comparison of these bounds with similar bounds for bounded distance decoding and majority decoding shows that recursive decoding can reduce the output error probability of the latter two algorithms by five or more orders of magnitude even on the short block length of PST.It is also proven that the error probability of recursive decoding can be exponentially reduced by eliminating one or a few information bits from the original RM code. Index Terms—Binarysymmetric channel, Chernoff bound, Plotkin construction, recursive decoding, Reed–Muller (RM) codes.
Error exponents for two soft decision decoding algorithms of Reed–Muller codes
"... Error exponents are studied for recursive and majority decoding algorithms of general ReedMuller codes RM(r; m) used on an AWGN channel. Both algorithms have low complexity order and are capable of correcting many error patterns beyond half the distance. Decoding consists of multiple consecutive s ..."
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Error exponents are studied for recursive and majority decoding algorithms of general ReedMuller codes RM(r; m) used on an AWGN channel. Both algorithms have low complexity order and are capable of correcting many error patterns beyond half the distance. Decoding consists of multiple consecutive steps, which repeatedly recalculate the input symbols, rst replacing them with their products and sums and then using softdecision majority voting. To de ne error exponents, we study the probabilities of the symbols obtained in these recursive recalculations. The end result are analytical upper bounds on the output error probability that hold for recursive and majority decoding algorithms of any code RM(r; m):
IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1 An Improved MajorityLogic Decoder Offering Massively Parallel Decoding for RealTime Control in Embedded Systems
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