Generalized Hamming weights of q-ary Reed-Muller codes

Heijnen, F.; Pellikaan, Ruud

doi:10.1109/18.651015

Cited by 131 publications

(140 citation statements)

References 36 publications

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“…The result concerning d(C(λ)) and d(C(δ)) is known as the order bound and comes from [15]. The result concerning d t (C(λ)) is from [13] and the result concerning d t (C(δ)) is from [12]. The results concerning E(λ) andẼ(λ) are new.…”

Section: Proposition 27 Letmentioning

confidence: 97%

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Evaluation codes from order domain theory

Andersen

Geil

2008

Finite Fields and Their Applications

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The celebrated Feng-Rao bound estimates the minimum distance of codes defined by means of their parity check matrices. From the Feng-Rao bound it is clear how to improve a large family of codes by leaving out certain rows in their parity check matrices. In this paper we derive a simple lower bound on the minimum distance of codes defined by means of their generator matrices. From our bound it is clear how to improve a large family of codes by adding certain rows to their generator matrices. The new bound is very much related to the Feng-Rao bound as well as to Shibuya and Sakaniwa's bound in [T. Shibuya, K. Sakaniwa, A dual of well-behaving type designed minimum distance, IEICE Trans. Fund. E84-A (2001) 647-652]. Our bound is easily extended to deal with any generalized Hamming weights. We interpret our methods into the setting of order domain theory. In this way we fill in an obvious gap in the theory of order domains.

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Section: Proposition 27 Letmentioning

confidence: 97%

“…From (23) and (24) is clear that our new bound gives exactly the same estimates for the generalized Reed-Muller codes as does the Feng-Rao bound. In [13] it was shown that the Feng-Rao bound is tight in the case of the generalized Hamming weights of generalized Reed-Muller codes. It follows immediately that so is our new bound.…”

Section: Example 51 Letmentioning

confidence: 98%

Evaluation codes from order domain theory

Andersen

Geil

2008

Finite Fields and Their Applications

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“…For the generalized Hamming weights of one-point codes there is a generalization of the order bound based also on the associated Weierstrass semigroups. We will not discuss this topic here but the reader interested in it can see [34,22].…”

Section: The ν Sequence and The Minimum Distance Of Classical Codesmentioning

confidence: 99%

Numerical Semigroups and Codes

Bras-Amorós¹

2013

Algebraic Geometry Modeling in Information Theory

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A numerical semigroup is a subset of N containing 0, closed under addition and with finite complement in N. An important example of numerical semigroup is given by the Weierstrass semigroup at one point of a curve. In the theory of algebraic geometry codes, Weierstrass semigroups are crucial for defining bounds on the minimum distance as well as for defining improvements on the dimension of codes. We present these applications and some theoretical problems related to classification, characterization and counting of numerical semigroups.Notice that a symmetric numerical semigroup can not be pseudo-symmetric. Next lemma as well as its proof is analogous to Lemma 10. Lemma 12.A numerical semigroup Λ with odd conductor c is pseudo-symmetric if and only if for any non-negative integer i different from (c − 1)/2, if i is a gap, then c − 1 − i is a non-gap. 9

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“…Finally Barbero and Munuera described all generalized Hamming weights for an arbitrary one-point code on a Hermitian curve [1]. Barbero and Munuera's main tool was the order bound on the generalized Hamming weights obtained by Heijnen and Pellikaan [5], which works only for one-point codes among the geometric Goppa codes. Namely, the theory is not applicable in the two-point case.…”

Section: Introductionmentioning

confidence: 99%