List Decoding of&amp;lt;tex&amp;gt;$q$&amp;lt;/tex&amp;gt;-ary Reed–Muller Codes

Pellikaan, Ruud; Wu, Xin-Wen

doi:10.1109/tit.2004.825043

Cited by 62 publications

(62 citation statements)

References 15 publications

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“…First, let us recall some basic facts about cyclic generalized Reed-Muller codes, see [9,10,53,75] for details. Let L m (ν) denote the subspace of F q [x 1 , .…”

Section: It Remains To Show That Swt((cmentioning

confidence: 99%

Nonbinary Stabilizer Codes Over Finite Fields

Ketkar

Klappenecker

Kumar

et al. 2006

IEEE Trans. Inform. Theory

549

560

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One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been derived as binary stabilizer codes. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over F q 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. This paper derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper.

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“…First, let us recall some basic facts about cyclic generalized Reed-Muller codes, see [9,10,53,75] for details. Let L m (ν) denote the subspace of F q [x 1 , .…”

Section: It Remains To Show That Swt((cmentioning

confidence: 99%

Nonbinary Stabilizer Codes Over Finite Fields

Ketkar

Klappenecker

Kumar

et al. 2006

IEEE Trans. Inform. Theory

549

560

View full text Add to dashboard Cite

show abstract

“…Two important results in this area include the Gurswami-Sudan list-decoder for Reed-Solomon codes that can correct η = 1 − d/q fraction of errors [15], and the Sudan-Trevisan-Vadhan local list-decoder which corrects η = 1 − 2d/q fraction of errors [27]. Pellikaan and Wu give a reduction from Reed-Muller decoding to Reed-Solomon decoding when d < q [23], which allows list-decoding up to η = 1 − d/q in the global setting. Other work addressing the large field setting includes [3,12].…”

Section: Reed-muller Codesmentioning

confidence: 99%

List-decoding reed-muller codes over small fields

Gopalan

Klivans

Zuckerman

2008

Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing

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We present the first local list-decoding algorithm for the r th order Reed-Muller code RM(r, m) over F2 for r ≥ 2. Given an oracle for a received word R : F m 2 → F2, our randomized local list-decoding algorithm produces a list containing all degree r polynomials within relative distance (2 −r − ε) from R for any ε > 0 in time poly(m r , ε −r ). The list size could be exponential in m at radius 2 −r , so our bound is optimal in the local setting. Since RM(r, m) has relative distance 2 −r , our algorithm beats the Johnson bound for r ≥ 2. In the setting where we are allowed running-time polynomial in the block-length, we show that list-decoding is possible up to even larger radii, beyond the minimum distance. We give a deterministic list-decoder that works at error rate below J(2 1−r ), where J(δ) denotes the Johnson radius for minimum distance δ. This shows that RM(2, m) codes are list-decodable up to radius η for any constant η < 1 2 in time polynomial in the block-length.Over small fields Fq, we present list-decoding algorithms in both the global and local settings that work up to the list-decoding radius. We conjecture that the list-decoding radius approaches the minimum distance (like over F2), and prove this holds true when the degree is divisible by q − 1.

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“…Thus the classical Reed-Solomon decoding algorithms can then be used, and this leads to an algorithm for the multivariate setting decoding up to half the minimum distance. In fact, Pellikaan-Wu [15] observed that this connection allows one to decode multivariate polynomial codes beyond half the minimum distance too, provided S is special in the above sense.…”

Section: Related Workmentioning

confidence: 98%

“…The problem of list-decoding Reed-Muller codes over general product sets S m up to the Johnson radius is a very interesting open problem left open by our work. Again, when S is algebraically special, it is known how to solve this problem [15]. Generalizing our approach seems to require progress on another very interesting open problem, that of list-decoding Reed-Solomon concatenated codes.…”

Section: Open Problemsmentioning

confidence: 99%