New generalizations of the Reed-Muller codes--I: Primitive codes

Kasami, Tadao; Lin, Shu; Peterson, W. W.

doi:10.1109/tit.1968.1054127

Cited by 210 publications

(133 citation statements)

References 2 publications

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“…The minimum distance d was established four decades ago in [6]. Soon after in [3] the polynomials producing codewords of weight d were shown all to be products of linear factors.…”

Section: Introductionmentioning

confidence: 99%

On the second weight of generalized Reed-Muller codes

Geil

2008

Des. Codes Cryptogr.

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Not much is known about the weight distribution of the generalized Reed-Muller code RM q (s, m) when q > 2, s > 2 and m ≥ 2 . Even the second weight is only known for values of s being smaller than or equal to q/2. In this paper we establish the second weight for values of s being smaller than q. For s greater than (m − 1)(q − 1) we then find the first s + 1 − (m − 1)(q − 1) weights. For the case m = 2 the second weight is now known for all values of s. The results are derived mainly by using Gröbner basis theoretical methods.

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“…The minimum distance d was established four decades ago in [6]. Soon after in [3] the polynomials producing codewords of weight d were shown all to be products of linear factors.…”

Section: Introductionmentioning

confidence: 99%

On the second weight of generalized Reed-Muller codes

Geil

2008

Des. Codes Cryptogr.

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“…This statement is well-known [7,18], but it follows also from Theorem 5.10. Let α i = 0 for all i < m − θ, α m−θ = µ, and α i = q − 1 for all…”

Section: Definition 54mentioning

confidence: 65%

“…The rth generalized Hamming weight (GHW) or rth minimum support weight of a linear code C is defined by The RM codes were defined in [32,28] for the binary case and this was generalized to arbitrary q in [7,18,43]. Generalized Hamming weights were introduced in the study [15,24] on the weight distribution of codes over extensions of F q .…”

Section: Introductionmentioning

confidence: 99%

Generalized Hamming weights of q-ary Reed-Muller codes

Heijnen¹,

Pellikaan²

Proceedings of IEEE International Symposium on Information Theory

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“…We can now recall the definition of Reed-Muller codes over F q , also called Generalized ReedMuller codes [6]:…”

Section: A Reed-muller Codes Over F Q and Information Setsmentioning

confidence: 99%

Information sets of Multiplicity codes

Augot

Levy-dit-Vehel

Ngo

2015

2015 IEEE International Symposium on Information Theory (ISIT)

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Abstract-We here provide a method for systematic encoding of the Multiplicity codes introduced by Kopparty, Saraf and Yekhanin in 2011. The construction is built on an idea of Kopparty. We properly define information sets for these codes and give detailed proofs of the validity of Kopparty's construction, that use generating functions. We also give a complexity estimate of the associated encoding algorithm.

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New generalizations of the Reed-Muller codes--I: Primitive codes

Cited by 210 publications

References 2 publications

On the second weight of generalized Reed-Muller codes

On the second weight of generalized Reed-Muller codes

Generalized Hamming weights of q-ary Reed-Muller codes

Information sets of Multiplicity codes

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