On the dual minimum distance and minimum weight of codes from a quotient of the Hermitian curve

Ballico, Edoardo; Ravagnani, Alberto

doi:10.1007/s00200-013-0206-z

Cited by 2 publications

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“…Take q = 9 and m = 5 for example. It follows from [30] that the number of rational places of the curve y 5 = x 9 + x with genus g = 16 is N = 1 + q(1 + (q − 1)m) = 370. From Corollary 14, we can get all the pure gaps at (P 1 , P ∞ ), which are showed in Figure 2.…”

Section: Examples Of Codes In Kummer Extensionsmentioning

confidence: 99%

Multi-point codes over Kummer extensions

Hu¹,

Yang

2017

Des. Codes Cryptogr.

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This paper is concerned with the construction of algebraic geometric codes defined from Kummer extensions. It plays a significant role in the study of such codes to describe bases for the Riemann-Roch spaces associated with totally ramified places. Along this line, we give an explicit characterization of Weierstrass semigroups and pure gaps. Additionally, we determine the floor of a certain type of divisor introduced by Maharaj, Matthews and Pirsic. Finally, we apply these results to find multi-point codes with good parameters. As one of the examples, a presented code with parameters [254, 228,16] over F64 yields a new record. Index TermsAlgebraic geometric codes, Kummer extension, Weierstrass semigroup, Weierstrass pure gap. I. INTRODUCTIONT HE algebraic geometric (AG) codes were introduced by V.D. Goppa [1], which were defined as the image of the Riemann-Roch space by the evaluation at several rational places. Since then, the study of AG codes becomes an important instrument in theory and practice. The famous Tsfasman-Vlǎduţ-Zink theorem says that the parameters of the AG codes associated with asymptotically good towers are better that the Gilbert-Varshamov bound [2], [3]. Pellikaan, Shen and van Wee [4] showed that any arbitrary linear code is in fact an AG code.Given an AG code of fixed length, the first task is to determine its parameters: dimension and minimum distance. In order to determine the dimension and construct the generator matrix, it is necessary to calculate the related Riemann-Roch space. By means of the Riemann-Roch theorem, one obtains a non-trivial lower bound, Goppa bound, for the minimum distance in a very general setting [5]. Garcia, Kim and Lax improved the Goppa bound using arithmetical structure of the Weierstrass gaps at one place in [6], [7]. Homma and Kim [8] introduced the concept of pure gaps and demonstrated a similar result for a divisor concerning a pair of places. And this was generalized to several places by Carvalho and Torres in [9]. Maharaj, Matthews and Pirsic [10], [11] extended this construction by introducing the notion of the floor of a divisor and obtained improved bounds on the parameters of AG codes. Codes over specific Kummer extensions were well-studied in the literature. For instance, Hermitian curves play an important role in coding theory due to their efficient encoding and decoding algorithms. Almost all of the known maximal curves arise from Hermitian curves. See [12], [13] and the references therein. Many authors examined one-point codes from Hermitian curves and developed efficient methods to decode them [5], [14], [15], [16]. The minimum distance of Hermitian two-point codes had been first determined by Homma and Kim [17], [18], [19], [20]. In [10], Maharaj and Matthews determined explicit bases for the Riemann-Roch space of a divisor of the form rP ∞ + E, where the support of E lies on a line. This allowed them to give an explicit formula for the floor of such a divisor. In [21], Geil considered codes from norm-trace curves and determined the true minimum...

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Section: Examples Of Codes In Kummer Extensionsmentioning

confidence: 99%