The equivalence of L-step orthogonalization and a Reed decoding procedure (Corresp.)

Gore, W.

doi:10.1109/tit.1969.1054241

Cited by 12 publications

(3 citation statements)

References 3 publications

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“…Given j, let P and Q = 1 0 P be the probabilities that the jth check x2X u x gives the valuesâ j = 1 andâ j = 0, respectively. Obviously,â j = 1, if the check includes odd number of perturbed symbols (that is "1's" (5) Similarly to [4], we expand the expression (q 0 p) 2 and see that…”

Section: =2mentioning

confidence: 99%

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Soft-decision majority decoding of Reed-Muller codes

Dumer

Krichevskiy

2000

IEEE Trans. Inform. Theory

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Abstract-We present a new soft-decision majority decoding algorithm for Reed-Muller codes RM (). First, the reliabilities of 2 transmitted symbols are recalculated into the reliabilities of 2 parity checks that represent each information bit. In turn, information bits are obtained by the weighted majority that gives more weight to more reliable parity checks. It is proven that for long low-rate codes RM ( ), our soft-decision algorithm outperforms its conventional hard-decision counterpart by 10 log ( 2) 2 dB at any given output error probability. For fixed code rate and , our algorithm increases almost 2 times the correcting capability of soft-decision bounded distance decoding.

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Section: =2mentioning

confidence: 99%

“…Reed-Muller (RM) codes RM (r; m) [10] and code distance d = 2 m0r : Their majority decoding has been introduced in the seminal paper [13] followed by numerous developments (see [1], [5], [7], [11], [12], [14], [17], and references therein). The algorithm provides bounded distance decoding with complexity order of nk or less.…”

Section: Introductionmentioning

confidence: 99%

Soft-decision majority decoding of Reed-Muller codes

Dumer

Krichevskiy

2000

IEEE Trans. Inform. Theory

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“…Despite relatively bad code distance, RM codes have been considered in numerous publications thanks to efficient decoding procedures. Majority decoding developed in [9] (see also [2], [4], [8], and [10]) has complexity order at most nk and corrects all error patterns of weight below d/2. It is also known [5] that majority decoding corrects many error patterns of higher weights and can be further improved for soft-decision channels [13].…”

Section: Introductionmentioning

confidence: 99%

Recursive decoding of Reed-Muller codes

Dumer

Shabunov²

2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060)

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New soft-and hard decision decoding algorithms are presented for general Reed-Muller codes m r of length 2 m and distance 2 m−r . We use Plotkin (u, u + v) construction and decompose code m r onto subblocks u ∈ m−1 r and v ∈ m−1 r−1. In decoding, we first try to find a subblock v from the better protected code and then proceed with the block u. The likelihoods of the received symbols are recalculated in a way similar to belief propagation. Thus, decoding is relegated to the two constituent codes. We repeat this recursion and execute decoding only at the end nodes j 1 and j j−1 . The overall complexity has low order of n log n. It is shown that this decoding substantially outperforms other algorithms of polynomial complexity known for RM codes. In particular, for medium and high code rates, the algorithm corrects most error patterns of weight d ln d/2.

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Some Combinatorial Aspects of Coding Theory

Goethals¹

1973

A Survey of Combinatorial Theory

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The equivalence of L-step orthogonalization and a Reed decoding procedure (Corresp.)

Cited by 12 publications

References 3 publications

Soft-decision majority decoding of Reed-Muller codes

Soft-decision majority decoding of Reed-Muller codes

Recursive decoding of Reed-Muller codes

Some Combinatorial Aspects of Coding Theory

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