Algebraic geometric codes with applications

The theory of linear error-correcting codes from algebraic geometric curves (algebraic geometric (AG) codes or geometric Goppa codes) has been well-developed since the work of Goppa and Tsfasman, Vladut, and Zink in 1981–1982. In this paper we introduce to readers some recent progress in algebraic geometric codes and their applications in quantum error-correcting codes, secure multi-party computation and the construction of good binary codes.      

On the duals of geometric Goppa codes from norm-trace curves

In this paper we study the dual codes of a wide family of evaluation codes on norm-trace curves. We explicitly find out their minimum distance and give a lower bound for the number of their minimum-weight codewords. A general geometric approach is performed and applied to study in particular the dual codes of one-point and two-point codes arising from norm-trace curves through Goppaʼs construction, providing in many cases their minimum distance and some bounds on the number of their minimum-weight codewords. The results are obtained by showing that the supports of the minimum-weight codewords of the studied codes obey some precise geometric laws as zero-dimensional subschemes of the projective plane. Finally, the dimension of some classical two-point Goppa codes on norm-trace curves is explicitely computed.     

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.     

On binary self-dual codes of lengths 60, 62, 64 and 66 having an automorphism of order 9

A method for constructing binary self-dual codes having an automorphism of order p ² for an odd prime p is presented in (S. Bouyuklieva et al. IEEE. Trans. Inform. Theory, 51, 3678–3686, 2005). Using this method, we investigate the optimal self-dual codes of lengths 60 ≤ n ≤ 66 having an automorphism of order 9 with six 9-cycles, t cycles of length 3 and f fixed points. We classify all self-dual [60,30,12] and [62,31,12] codes possessing such an automorphism, and we construct many doubly-even [64,32,12] and singly-even [66,33,12] codes. Some of the constructed codes of lengths 62 and 66 are with weight enumerators for which the existence of codes was not known until now.      

A generalized floor bound for the minimum distance of geometric Goppa codes

We prove a new bound for the minimum distance of geometric Goppa codes that generalizes two previous improved bounds. We include examples of the bound applied to one- and two-point codes over certain Suzuki and Hermitian curves.     

The New Minimum Distance Bounds of Goppa Codes and Their Decoding

A couple of new lower bounds of the minimum distance of Goppa codes is derived, using an extended field code for a Goppa code which contains the Goppa code as its subfield-subcode. Also presented are procedures for both error-only and error-and-erasure decoding for Goppa codes up to the new lower bounds, based on the Berlekamp-Massey algorithm and the Feng-Tzeng multisequence shift-register synthesis algorithms which have been used for decoding cyclic codes up to the BCH and HT(Hartmann-Tzeng) bounds.     

Equality of geometric Goppa codes and equivalence of divisors

We give necessary and sufficient conditions for two geometric Goppa codes C_L(D,G) and C_L(D,H) to be the same. As an application we characterize self-dual geometric Goppa codes.     

Abelian groups admitting fixed-point-free automorphisms of order qn

For a wide class of Abelian groups, necessary and sufficient conditions under which a group admits an automorphism of order qⁿ are found; we also present necessary and sufficient conditions under which a group admits an automorphism ϕ of order qⁿ such that ϕ^qm is a fixed-point-free automorphism for some m < n. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 239–249, February, 1977.     

GOPPA CODES FROMARTIN-SCHREIER FUNCTION FIELDS

ＧＯＰＰＡＣＯＤＥＳＦＲＯＭＡＲＴＩＮ－ＳＣＨＲＥＩＥＲＦＵＮＣＴＩＯＮＦＩＥＬＤＳ￥ＨＡＮＷＥＮＢＡＯ（ＤｅｐａｔｍｅｌｌｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｉｃｈｕａｎＵｎｉｖｅｒｓｉｔｙｔＣｈｅｎｇｄｕ６１００６４，Ｓｉｃｈｕａｎ，Ｃｈｉｎａ．）Ａ...     

Weight distributions of geometric Goppa codes

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 -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.

     

McEliece Public Key Cryptosystems Using Algebraic-Geometric Codes

McEliece proposed a public-key cryptosystem based on algebraic codes, in particular binary classical Goppa codes. Actually, his scheme needs only a class of codes with a good decoding algorithm and with a huge number of inequivalent members with given parameters. In the present paper we look at various aspects of McEliece's scheme using the new and much larger class of R-ary algebraic-geometric Goppa codes.     

A Subclass of Binary Goppa Codes With Improved Estimation of the Code Dimension

A new bound for the dimension of binary Goppa codes belonging to a specific subclass is given. This bound improves the well-known lower bound for Goppa codes.     

Goppa codes over the p-adic integers and integers modulo pe

Goppa codes were defined by Valery D. Goppa in 1970. In 1978, Robert J. McEliece used this family of error-correcting codes in his cryptosystem, which has gained popularity in the last decade due to its resistance to attacks from quantum computers. In this paper, we present Goppa codes over the p-adic integers and integers modulo $p^e$. This allows the creation of chains of Goppa codes over different rings. We show some of their properties, such as parity-check matrices and minimum distance, and suggest their cryptographic application, following McEliece's scheme.     

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 ${\mathbb{F}}_{q^m}$ 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.     

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

Designs, Codes and Cryptography - In this paper, we construct a family of non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes. Moreover, we present a...     

Singly-Even Self-Dual Codes of Length 40

All singly-even self-dual [40,20,8] binary codes which have an automorphism of prime order are obtained up to equivalence. There are two inequivalent codes with an automorphism of order 7 and 37 inequivalent codes with an automorphism of order 5. These codes have highest possible minimal distance and some of them are the first known codes with weight enumerators prescribed by Conway and Sloane.     

GENERALIZED GEOMETRIC GOPPA CODES

ABSTRACT

Conventional geometric Goppa codes are defined in terms of functions of an algebraic function field associated with a divisor evaluated in places of degree 1. The generalization that will be treated here allows evaluations in places of arbitrary degree. With the appropriate inner product, the dual of the code can be defined and described in terms of Weil differentials similarly to conventional geometric Goppa codes. A decoding algorithm is derived.     

Binary self-dual codes with automorphisms of order 23

The only example of a binary doubly-even self-dual [120,60,20] code was found in 2005 by Gaborit et al. (IEEE Trans Inform theory 51, 402–407 2005). In this work we present 25 new binary doubly-even self-dual [120,60,20] codes having an automorphism of order 23. Moreover we list 7 self-dual [116,58,18] codes, 30 singly-even self-dual [96,48,16] codes and 20 extremal self-dual [92,46,16] codes. All codes are new and present different weight enumerators.      

Hamming spaces and maximal self dual codes over GF(q), q=ODD

The group of automorphisms of a Hamming space is determined. Self dual codes over odd characteristic finite field with respect to bilinear forms are treated. Under the subgroup of the monomial group preserving the inner product, we classify the maximal self dual codes overGF(5) with respect to the inner product of dimension ≤8. The Hamming Weight distribution and the order of the automorphism of the code are given. Partially supported by NSERC A8460 and Scarborough College.     

Asymptotically good families of subfield subcodes of geometric Goppa codes

We show that subfield subcodes of certain geometric Goppa codes meet the Gilbert-Varshamov bound. A very special case of our theorem is the well known fact that classical Goppa codes meet the Gilbert-Varshamov bound.