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# Automorphism group of generalized Hermite code weight distribution and weight is the minimum number of codewords of Hamming weight

Author: ZhuJian
Tutor: CaiChuanRen
School: Yangzhou University
Course: Basic mathematics
Keywords: generalized Hermitian code Hamming distance
CLC: O157.4
Type: Master's thesis
Year: 2007
Quote: 0

### Abstract

 The main theoretical foundation to research algebraic geometry code is algebraic geometry. The research on encoding the property of the codes should apply the concept and related theorem of algebra and geometry, especially the three most important theorem of algebraic geometry: Riemann - Roch theorem, Hasse - Weil theorem and Bezout theorem. Especially, Riemann - Roch theorem is the core tools to study algebraic geometry code. In this paper,we can see its application in Theorem 3 and Theorem 4.Under the assignment of different ways, algebraic geometry code can be divided into two categories: the geometry RS code and geometric Goppa code. It can be proved that geometric RS code and geometric Goppa code are dual codes for each other, and C(D,G) = C ? ( D , W + D ? G), where W is a canonical divisor. As for the dual codes, their weight distribution are intrinsically linked, as the Macwilliams constant equation stated. Therefore, generally speaking, we can study the geometric RS code only.Algebraic geometry code has very good properties, it is generated by the algebraic curves on the limited domain, and the properties of the codes are determined by the properties of algebraic curves of nature. When searching or studying on some very beautiful algebraic curves, we may also get good properties of algebraic geometry code. For the Hermitian curve y q + y = x q+1 on Fq 2 , we can see that the left part is a trace mapping, and the right part is mode mapping. The number of rational points on the curve reached the Hasse - Weil bond, so the codes generated by this curve posit very good properties.In this paper we study the generalized Hermitian curve y q + y = xs, where s≠q+ 1.. It also reached Hasse - Weil bond, and the properties codes are very similar to Hermitian. For example, there exists affine automorphism group on the Hermitian curve, and we can find similar affine automorphism group through similar approaches on generalized Hermitian curve. However, the curves which exist affine automorphism group are very few. In [7], the authors defined the function of the affine automorphism group to Hermitian code, and the function of the group to the ser can be also moved to the general case. In [8], the authors generized the affine automorphism group on the Hermitian curve, and made a comprehensive study of all possible automorphism group on the Hermitian code. The affine automorphism group for curves is the subgroups of the automorphism group of the code.In [7], through the effect from the affine automorphism group to codes, they can get one conclusion for on the weight distribution of the codes. When ( , q ) = 1, the total number a of the codes weighted satisfied a≡0 mod( q 3 ( q 2? 1)). Then for the generalized case, we can get a similar conclusion. Particularly, it should be noted that when s = 1 for the curve y q+ y = x, its affine automorphism group will arise variation, and the rankof the group will be relatively large. Then correspondingly, for the majority of value , the total number a of the codes weighted will be divisible by a relatively large number.For the minimum distance of codes, minimum distance of Hermitian code C ( D , mP∞) is specific in [10]. [9] proposed the concept of a draft Hermitian curve. Such curves are very extensive and the generalized Hermitian curve is its special case. In this paper, we studied the draft Hermitian code C ( D , mP∞). When m is set to some values, the minimum distance will achieve the low bound, However, no specific value for m is given as it is very complicated for specific cases and the content of Hermitian curve is too wide. Therefore in this paper we give some conditions for m to meet in order to achieve the low bound for the minimum distance for the generalized Hermitian code C ( D , mP∞). Also for the number of codes with minimum Hamming weight, we proposed one draft numerical estimate for the upper bound of that number. We have following three directions for the future research.1 To identify all the automorphism groups for generalized Hermitian code C ( D , mP∞).2 To give the exact weight distribution for the generalized Hermitian.3 For all values m , to calculate the minimum distance for the generalized Hermitian code C ( D , mP∞).This is of great significance for three directions.

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