Geometric approach to higher weights

“…For more on this relation, one may refer to [2,4,19]. We now derive a useful consequence of Wei duality and monotonicity of higher weights of a linear code.…”

Section: Projective Reed-muller Codesmentioning

confidence: 99%

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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“…It does suffice however to consider finitely many constant field extensions. The relations between weight distributions for different constant field extensions appear to be important for the study of generalized Hamming weights [33], [27].…”

Section: Theorem 95 For a Code C With Zeta Function Z(t ) Letmentioning

confidence: 99%

Weight distributions of geometric Goppa codes

Duursma

1999

Trans. Amer. Math. Soc.

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Abstract. The in general hard problem of computing weight distributions of linear codes is considered for the special class of algebraic-geometric codes, defined by Goppa in the early eighties. Known results restrict to codes from elliptic curves. We obtain results for curves of higher genus by expressing the weight distributions in terms of L-series. The results include general properties of weight distributions, a method to describe and compute weight distributions, and worked out examples for curves of genus two and three.

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Geometric approach to higher weights

Cited by 170 publications

References 25 publications

Asymptotically good quantum codes

Asymptotically good quantum codes

Arithmetic, Geometry, Cryptography and Coding Theory

Weight distributions of geometric Goppa codes

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