Generalized Hamming weights of q-ary Reed-Muller codes

Heijnen, F.; Pellikaan, R.

doi:10.1109/isit.1997.613295

Cited by 40 publications

(81 citation statements)

References 26 publications

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“…What is stated above are, in fact, special cases of the results of Heijnen and Pellikaan [10], which deals, more generally, with the case r ≤ m+d d…”

Section: Introductionmentioning

confidence: 91%

“…The theorem of Heijnen-Pellikaan is intimately related to the determination of higher weights (also known as, generalized Hamming weights) of Reed-Muller codes RM q (d, m) and in fact, that was the original motivation of [10]. In a similar manner, the TBC is closely related to determination of higher weights of projective ReedMuller codes PRM q (d, m).…”

Section: Introductionmentioning

confidence: 98%

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Arithmetic, Geometry, Cryptography and Coding Theory

Kohel

Rolland

2017

Contemporary Mathematics

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Abstract. Tsfasman-Boguslavsky Conjecture predicts the maximum number of zeros that a system of linearly independent homogeneous polynomials of the same positive degree with coefficients in a finite field can have in the corresponding projective space. We give a self-contained proof to show that this conjecture holds in the affirmative in the case of systems of three homogeneous polynomials, and also to show that the conjecture is false in the case of five quadrics in the 3-dimensional projective space over a finite field. Connections between the Tsfasman-Boguslavsky Conjecture and the determination of generalized Hamming weights of projective Reed-Muller codes are outlined and these are also exploited to show that this conjecture holds in the affirmative in the case of systems of a "large" number of three homogeneous polynomials, and to deduce the counterexample of 5 quadrics. An application to the nonexistence of lines in certain Veronese varieties over finite fields is also included.

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“…What is stated above are, in fact, special cases of the results of Heijnen and Pellikaan [10], which deals, more generally, with the case r ≤ m+d d…”

Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 98%

Arithmetic, Geometry, Cryptography and Coding Theory

Kohel

Rolland

2017

Contemporary Mathematics

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“…In the following section a sketch will be given how this gives rise to a sequence of codes C(l) and a bound on the minimum distance of these codes following the ideas of [5,6,7,11,12,13,19,20,22,23]. This was generalized to a bound on the generalized Hamming weights by [10].…”

Section: The Index Of Speciality I(g) Of a Divisor G Is A Nonnegativementioning

confidence: 99%

“…, X m ] to F n , since f g(P i ) = f (P i )g(P i ) for all polynomials f and g, and all i. The map ev P is surjective [3,10]. Suppose that I is an ideal in the ring F[X 1 , .…”

Section: A Bound On the Minimum Distancementioning

confidence: 99%

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On the existence of order functions

Pellikaan

2001

Journal of Statistical Planning and Inference

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On the Number of Solutions of Polynomial Systems

Boguslavsky

1997

Finite Fields and Their Applications

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We consider systems of homogenous polynomial equations of degree d in a projective space ‫ސ‬ m over a finite field ‫ކ‬ q . We attempt to determine the maximum possible number of solutions of such systems. The complete answer for the case r ϭ 2, d Ͻ q Ϫ 1 is given, as well as new conjectures about the general case. We also prove a bound on the number of points of an algebraic set of given codimension and degree. We also discuss an application of our results to coding theory, namely to the problem of computing generalized Hamming weights for q-ary projective Reed-Muller codes.

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Generalized Hamming weights of q-ary Reed-Muller codes

Cited by 40 publications

References 26 publications

Arithmetic, Geometry, Cryptography and Coding Theory

Arithmetic, Geometry, Cryptography and Coding Theory

On the existence of order functions

On the Number of Solutions of Polynomial Systems

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