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

Dumer, Ilya; Kabatiansky, Grigory; Tavernier, Cédric

doi:10.1109/isit.2006.261690

Cited by 24 publications

(8 citation statements)

References 7 publications

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“…We remark that the complexity in [7] was in O(n log 2 (n)/θ 2 ) for the same decoding radius. Hence, for θ ≤ 1/ log 2 (n), we improve the previous algorithm.…”

Section: Conjecturementioning

confidence: 89%

“…Subsequently, Goldreich et al [4] have generalized this algorithm over a finite field and suggested an extension for any order over a large finite field. Then, deterministic algorithms which work for any order have been suggested by Pellikaan and Wu in [5], and subsequently by Kabatiansky and Tavernier in [6] for the second order and by Dumer et al [7] for any order. This last algorithm considerably improves the algorithm of [5] by correcting up to the Johnson bound with an almost linear complexity.…”

mentioning

confidence: 99%

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An improved list decoding algorithm for the second order Reed–Muller codes and its applications

Fourquet

Tavernier

2008

Des. Codes Cryptogr.

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We propose an algorithm which is an improved version of the KabatianskyTavernier list decoding algorithm for the second order Reed-Muller code RM(2, m), of length n = 2 m , and we analyse its theoretical and practical complexity. This improvement allows a better theoretical complexity. Moreover, we conjecture another complexity which corresponds to the results of our simulations. This algorithm has the strong property of being deterministic and this fact drives us to consider some applications, like determining some lower bounds concerning the covering radius of the RM(2, m) code. AMS Classifications94B05 · 94B35 · 94B65 · 94A60 IntroductionIntroduced by Elias [1] 50 years ago, the concept of list decoding has been recently revived thanks to Sudan's discovery [2] of efficient list decoding algorithms for Reed-Solomon (RS) codes. Despite that there are similarities between Reed-Solomon and Reed-Muller (RM) codes, no efficient list decoding algorithm for RM codes was known until very recently. This article is a rewritten and completed version of Fourquet R., Tavernier C.: List decoding of second order Reed-Muller codes and its covering radius implications, Workshop on Coding and Cryptography 2007, pp. 147-156. R. Fourquet (B)

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“…We remark that the complexity in [7] was in O(n log 2 (n)/θ 2 ) for the same decoding radius. Hence, for θ ≤ 1/ log 2 (n), we improve the previous algorithm.…”

Section: Conjecturementioning

confidence: 89%

mentioning

confidence: 99%

An improved list decoding algorithm for the second order Reed–Muller codes and its applications

Fourquet

Tavernier

2008

Des. Codes Cryptogr.

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“…Even the exact values of second-order nonlinearity is known only for a few specific functions and for functions in small numbers of variables [13][14][15]23,24]. To say the least, up to now, proving a tight lower bound on the second-order nonlinearity is also a hard task except for a few special cases.…”

Section: Introductionmentioning

confidence: 99%

On the second-order nonlinearities of some bent functions

Tang

Carlet

Tang

2013

Information Sciences

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“…In [12,13,17] list decoding algorithms for higher order Reed-Muller codes are employed to compute second-order nonlinearities. These algorithms are successful for n ≤ 11 and for n ≤ 13 for some particular functions.…”

Section: Introductionmentioning

confidence: 99%

Third-order nonlinearities of a subclass of Kasami functions

Gode

Gangopadhyay

2010

Cryptogr. Commun.

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The rth-order nonlinearity, where r ≥ 1, of an n-variable Boolean function f , denoted by nl r ( f ), is defined as the minimum Hamming distance of f from all n-variable Boolean functions of degrees at most r. In this paper we obtain a lower bound of the third-order nonlinearities of Kasami functions of the form Tr n 1 (μx 57 ). It is demonstrated that for large values of n the lower bound of the third-order nonlinearities of the functions of this form is larger than the general lower bound obtained by Carlet (IEEE Trans Inf Theory 54 (3):1262-1272, 2008) for Kasami functions. Further we show that our result along with the computational results obtained by Fourquet and Tavernier (Designs Codes Cryptogr 49:323-340, 2008) provide us an estimate of the nonlinearity profiles of these functions for n = 7, 8, 10.

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

Cited by 24 publications

References 7 publications

An improved list decoding algorithm for the second order Reed–Muller codes and its applications

An improved list decoding algorithm for the second order Reed–Muller codes and its applications

On the second-order nonlinearities of some bent functions

Third-order nonlinearities of a subclass of Kasami functions

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