Multi-point codes over Kummer extensions

Abstract

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...

Pure gaps on curves with many rational places

“…The majority of maximal curves and curves with many rational places has a plane model of Kummer-type. For those curves with affine equation given by y m = f (x) λ where m ≥ 2, λ ≥ 1 and f (x) is a separable polynomial over F q , general results on gaps and pure gaps can be found in [1,5,18,29]. Applications to codes on particular curves such as the Giulietti-Korchmáros curve, the Garcia-Güneri-Stichtenoth curve, and quotients of the Hermitian curve can be found in [19,28,31].…”

On Weierstrass Gaps at Several Points

“…For instance, Bartoli, Quoos, and Zini [2] gave a criterion to find pure gaps at many places and presented families of pure gaps at several points on curves associated with Kummer extensions. In [12], Hu and Yang yielded a characterization of gaps and pure gaps at several points also on Kummer extensions. The same authors in [13] also furnished an arithmetic/combinatorial description of pure Weierstrass gaps at many totally ramified places on a quotient of the Hermitian curve.…”

“…To illustrate our approach, we present in this work a study of gaps and pure gaps at several points on X f,g . This family of curves contains the Kummer extensions and the quotient of the Hermitian curve, studied respectively in [2], [12], and [13]. This paper is organized as follows.…”

Multi-point codes from the GGS curves

“…Maharaj, Matthews and Pirsic [25] determined explicit bases for large classes of Riemann-Roch spaces of the Hermitian function field. Along this research line, Hu and Yang [19] gave other explicit bases for Riemann-Roch spaces of divisors over Kummer extensions, which makes it convenient to determine the pure gaps.…”

“…Bartoli, Montanucci and Zini [3] examined one-point AG codes from the GGS curves. Inspired by the above work and [8,19], here we will investigate multi-point AG codes arising from GGS curves. To be precise, an explicit basis for the corresponding Riemann-Roch space is determined by constructing a related set of lattice points.…”