埃爾米特碼

Abstract

1982年，Tsfasman， Vladut 和 Zink這三個數學家利用代數幾何碼序列發現了一個比有名的GV界更好的線性碼信息傳輸速度下界，並稱之爲TVZ界。此發現引起了編碼學家們對代數幾何碼的興趣。其中，基於埃爾米特函數域有著很好的特性， 埃爾米特碼被徹底的研究了。在文獻[8]中，典型的埃爾米特單點碼的最小距離已被一一算出了。 本論文將會討論幾種不同的埃爾米特碼，其中包括典型的埃爾米特單點碼、用大於一次的點造出的埃爾米特單點碼，以及埃爾米特多點碼。 論文的焦點會放在好的埃爾米特碼的構造方法。此外，本論文也會討論某些比典型埃爾米特單點碼更好的埃爾米特碼的存在性。最後，在例子中會展示一些埃爾米特碼的實際計算，以證明所造出來的碼確實會比典型的埃爾米特單點碼更好。

In 1982, Tsfasman, Vladut and Zink discovered a lower bound for the information rates of linear codes, known as the TVZ Bound, using sequences of algebraic geometry codes (AG codes). This discovery had brought the attention of coding theorists to AG codes. In this correspondence, the Hermitian codes has been study thoroughly, owing to the remarkable properties of Hermitian funciton fields. In fact, the true minimal distance of the classical one-point Hermitian codes has been dertermined in [8]. In this thesis, several families of Hermitian codes are discussed; namely, the classical one-point Hermitian codes, the one-point Hermitian codes supported by a place of degree higher than one, and the multple-point Hermitian codes. The focus of this thesis is laid on the consturction of some good Hermitian codes. Besides that, the existence of some Hermitian codes with parameters improved over the much-studied classical one-point Hermitian codes are also discussed. Last but not least, some concrete examples of Hermitian codes are constructed to show the improvement of parameters over the classical one-point Hermitian codes.