Extended Goppa Codes Research Articles

Generator Polynomials of Cyclic Expurgated or Extended Goppa Codes

Generator Polynomials of Cyclic Expurgated or Extended Goppa Codes

Galois hulls of special Goppa codes and related codes with application to EAQECCs

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over Fqm are GRS codes (resp., extended GRS codes) when m=1. In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs.

Construction of Expurgated and Extended Goppa Codes With Dihedral Automorphism Groups

In this paper, we determine all dihedral subgroups in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$PGL_{2}(\mathbb {F}_{q})$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q=2^{l}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$l$ </tex-math></inline-formula> is a positive integer; and construct binary expurgated and extended Goppa codes with dihedral automorphism groups. Moreover, it is easy to obtain binary quasi-cyclic expurgated or extended Goppa codes.

Non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

Non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

Using the Structure of Subfields in the Construction of Goppa Codes and Extended Goppa Codes

It is shown that for some location sets and some integral multiple power Galois fields that some new general subclasses of Goppa codes may be defined which have improved lower bounds to code dimension and minimum distance compared with ordinary Goppa codes. Some previously published results are shown to be particular cases of these general subclasses of codes. A new subclass of reversible Goppa codes is also presented. Examples of code construction for these subclasses, demonstrating the improved code parameters, are presented for both non binary and binary codes.

A Generalized Construction of Extended Goppa Codes

We present a generalized construction of extended length Goppa codes. Using this construction, we obtain 71 new codes in finite field F q for q = 4, 7, 8, 9 with better minimum distance than the previously known codes with the same length and dimension.

New binary codes from extended Goppa codes

An efficient construction of extended length Goppa codes is presented. The construction yields four new binary codes [153, 71, 25], [151, 70, 25], [160, 70, 27], and [158, 69, 27]. The minimum distances are larger than those of the best previously known linear codes of the same length and dimension.

New Bounds on the Distance Distribution of Extended Goppa Codes

We derive new estimates for the error term in the binomial approximation to the distance distribution of extended Goppa codes. This is an improvement on the earlier bounds by Vladuts and Skorobogatov, and Levy and Litsyn.

On the Cyclicity of Goppa Codes, Parity-Check Subcodes of Goppa Codes, and Extended Goppa Codes

Classical Goppa codes are a special case of Alternant codes. First we prove that the parity-check subcodes of Goppa codes and the extended Goppa codes are both Alternant codes. Before this paper, all known cyclic Goppa codes were some particular BCH codes. Many families of Goppa codes with a cyclic extension have been found. All these cyclic codes are in fact Alternant codes associated to a cyclic Generalized Reed–Solomon code. In (1989, J. Combin. Theory Ser. A 51, 205–220) H. Stichtenoth determined all cyclic extended Goppa codes with this property. In a recent paper (T. P. Berger, 1999, in “Finite Fields: Theory, Applications and Algorithms (R. Mullin and G. Mullen, Eds.), pp. 143–154, Amer. Math. Soc., Providence), we used some semi-linear transformations on GRS codes to construct cyclic Alternant codes that are not associated to cyclic GRS codes. In this paper, we use these results to construct cyclic Goppa codes that are not BCH codes, new families of Goppa codes with a cyclic extension, and some families of non-cyclic Goppa codes with a cyclic parity-check subcode.

New classes of cyclic extended Goppa codes

We give a condition for a "classical" Goppa (1970) code to have a cyclic extension. This condition follows from the properties of the semilinear automorphism group of the corresponding generalized Reed-Solomon codes. Using this condition, we can construct not only all the known Goppa codes with a cyclic extension, but also new families.

Which extended goppa codes are cyclic?

This paper contains a partial solution of a problem of MacWilliams and Sloane concerning the cyclicity of an extended “classical” Goppa code. The main idea of the proof is to represent the code as a “geometric” Goppa code.over the rational function field and a result from (Stichtenoth, Automorphisms of geometric Goppa codes, [8]) which states that under certain assumptions all automorphisms of such geometric codes are induced by projective transformations.