On the true minimum distance of Hermitian codes

Yang, Kyeongcheol; Kumar, P. Vijay

doi:10.1007/bfb0087995

Cited by 106 publications

(92 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…En particular la distancia mínima de códigos unipuntuales Hermitianos fue establecida por Yang y Kumar en [18]. Nosotros utilizando los resultados de la sección anterior obtenemos una caracterización más práctica para determinar la distancia mínima de estos códigos.…”

Section: Códigos Unipuntuales Hermitianosunclassified

“…Note que esta caracterización del valor exacto de la distancia mínima de códigos Hermitianos es más simple que las conocidas en la literatura (ver [18] y [9]) y resalta el hecho de que la distancia mínima sólo difiere de la cota Goppa cuando m i = n − l w para toda laguna l w . Observe también que se puede construir un [64,53,8]-código sobre F 16 que es un nuevo récord, según las tablas MinT [13].…”

unclassified

See 1 more Smart Citation

Sobre la distancia mínima de códigos AG unipuntuales castillo

León

Pinzón

2012

ing.cienc.

View full text Add to dashboard Cite

MSC: 94.B27, 94.B65Resumen Presentamos una caracterización de la cota inferior d * para la distancia mínima de códigos algebraico-geométricos unipuntuales sobre curvas castillo. Calculamos explícitamente esta cota en el caso de un semigrupo de Weierstrass generado por dos elementos consecutivos. En particular, obtenemos una caracterización más simple del valor exacto de la distancia mínima para códigos Hermitianos.

show abstract

Section: Códigos Unipuntuales Hermitianosunclassified

unclassified

Sobre la distancia mínima de códigos AG unipuntuales castillo

León

Pinzón

2012

ing.cienc.

View full text Add to dashboard Cite

show abstract

“…Thus the codelength is governed by the number of rational points on the curve , and the dimension of the AG code is smaller than that of MDS code with the same and , by an amount equal to the genus of If is not in the range , the dimension may be higher than the value given by (107) [30].…”

Section: Comparison To Algebraic-geometry Codesmentioning

confidence: 93%

“…Fortunately, the true minimum distance of Hermitian codes has been exactly determined in [30]. It is also known that, with the recent decoding algorithm of Feng and Rao [6], we can decode Hermitian codes up to the true minimum distance [14].…”

Section: Comparison To Algebraic-geometry Codesmentioning

confidence: 99%

“…It is also known that, with the recent decoding algorithm of Feng and Rao [6], we can decode Hermitian codes up to the true minimum distance [14]. Therefore, for a fair comparison to SSRS codes, we use the true minimum distance of Hermitian codes from [30]. Let us try to compare the family of Hermitian codes of length over GF to their SSRS counterparts.…”

Section: Comparison To Algebraic-geometry Codesmentioning

confidence: 99%

See 1 more Smart Citation

Subspace subcodes of Reed-Solomon codes

Hattori

McEliece

Lin

Proceedings of 1994 IEEE International Symposium on Information Theory

View full text Add to dashboard Cite

Abstract-In this paper we introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed -dimensional vector subspace S of GF (2 m ): SSRS codes are constructed using properties of the Galois field GF (2 m ): They are not linear over the field GF (2 ), which does not come into play, but rather are Abelian group codes over S: However, they are linear over GF (2), and the symbolwise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that 1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF (2 ) is a subfield of GF (2 m ), it may not be the best -dimensional subspace for constructing SSRS codes; and 2) many high-rate SSRS codes have larger dimension than any previously known code with the same values of n; d; and q; including algebraicgeometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems.

show abstract

Explicit Computation of Generalized Hamming Weights for Some Algebraic Geometric Codes

Chen¹,

Luk

Yau

1998

Advances in Applied Mathematics

View full text Add to dashboard Cite

The generalized Hamming weights, introduced a few years ago by V. K. Wei, provide substantial information of codes and thus play a central role in coding theory. For algebraic geometric codes, there have been many works on their Ž . generalized Hamming weights or weight hierarchy . However, for lots of codes from Hermitian curves and the Klein quartic, some generalized Hamming weights still have not yet been found explicitly. In this paper, we first prove a general result Ž . Theorem 1.4 on the computation of generalized Hamming weights of geometric Goppa codes on plane curves, using the configuration of F -rational points on the q Ž . curves. Then we give the exact values Theorem 2.2 of the first and second generalized Hamming weights of some codes arising from the Klein quartic. Our Ž . main result Theorem 2.3 gives the exact values of the second and third generalized Hamming weights of certain codes from Hermitian curves. In the Appendix, a previous known result of Yang, Kumar, and Stichtenoth for Hermitian codes is shown to follow from Theorem 1.4. We also give the exact values of the first three generalized Hamming weights for Fermat codes. ᮊ

show abstract

On the true minimum distance of Hermitian codes

Cited by 106 publications

References 3 publications

Sobre la distancia mínima de códigos AG unipuntuales castillo

Sobre la distancia mínima de códigos AG unipuntuales castillo

Subspace subcodes of Reed-Solomon codes

Explicit Computation of Generalized Hamming Weights for Some Algebraic Geometric Codes

Contact Info

Product

Resources

About