环Z_（p_（k+1））上的（1+p~k）-循环码与Gray映射
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摘要
上世纪70年代起，Blake[1]和Speigel[2]等学者开始将纠错码的研究从有限域上转移到整数剩余类环Z_m上．90年代初，Forney等学者在[3]，Hammons等学者在[4]中证明了Kerdock码，Preparata码，Delsarte-Goethals码比同样长度，同样距离的线性码有更多的码字，这些非线性码实际上就是一些Z_4上的线性码在Gray映射下的像．1998年，Carlet在文献[6]中，通过Boolean函数在Z_(2~k)上定义了Gray映射，通过Gray映射将Z_(2~k)上的线性码映射成Z_2上的非线性码，得到了广义的Kerdock码和广义的Goethals码．Ling在文献[9]中进一步将Gray映射推广到环Z_(p~(k+1))上，给出了(1-p~k)-循环码的Gray像是Z_p上的准循环码，且通过建立(1-p~k)-循环码与一般循环码的一一对应，得到了环Z_(p~(k+1))上的循环码的Gray像等价于准循环码，且给出了它们的像为线性的充分条件．
本文继续对环Z_(p~(k+1))上的码展开研究，得到了以下主要结果．
在第二章，通过利用环Z_(p~(k+1))中的元素可以唯一写成p进制的形式，以及从Z_(p~(k+1))~n到Z_p~(p~kn)的Gray映射，我们给出了环Z_(p~(k+1))上的(1+p~k)-循环码的Gray像和一般循环码的Gray像以及负循环码的Gray像。
在第三章，我们给出了环Z_(p~(k+1))上长为n，(n，p)=1的常循环码的生成元．通过建立Z_(p~(k+1))上的循环码与(1+p~k)-循环码的一一对应给出了(1+p~k)-循环码的生成元．
在第四章，我们考虑环Z_(p~2)上的码．由于本文所说的循环码，常循环码，准循环码不一定是线性的．最后本文给出了环Z_(p~2)上(1+p)-循环码和循环码的Gray像是线性的充分条件．
Since 1970s, some researchers, such as Blake[1] and Speigel[2], began to discuss error-correcting codes over the rings Z_m of integers modulo m instead of studying codes over finite fields. In the beginning of 1990s, Forney etc[3] and Hammous etc[4] proved that some nonlinear codes, such as Kerdock code, Preparata code and Delsarte-Goethals code are the images of some linear codes over Z_4, where the map is the Gray map. Since these non-linear codes have more codewords than those linear codes with the same length and the same Hamming distance, the research interest in codes over finite rings has grown rapidly. In 1998, Carlet[6] defined Gray map over Z_(2~k) by using Boolean function, then mapped the linear codes over Z_(2~k) to nonlinear codes over Z_2 by the Gray map in this paper, and finally obtained generalized Kerdock code and generalized Goethals code. Ling[9] extended the definition of Gray map to Z_(p~(k+1)) and proved that the Gray image of (1-p~k)- cyclic codes are quasi-cyclic codes over Z_p. In [9], they also proved that cyclic codes over the rings Z_(p~(k+1)) are equivalent to quasi-cyclic codes by formulating a one-to-one correspondence between (1-p~k)-cyclic codes and general cyclic codes, they also provided a sufficient condition for the images of codes over Z_(p~(k+1)) to be linear.
In this thesis, we shall continue the study on codes over Z_(p~(k+1)). We obtain the following results.
In chapter 2, by using the Gray map (see [9]) from Z_(p~(k+1))~n to Z_p~(p~kn), and the unique p-adic expression of each element in Z_(p~(k+1)), the Gray images of (1+p~k)- cyclic codes of length n over the rings Z_(p~(k+1)), general cyclic codes and negacyclic codes are obtained, where (n,p) = 1.
In chapter 3, when (n,p) = 1, the generator of constacyclic codes with length n over Z_(p~(k+1)) are obtained. By using the one-to-one correspondence between cyclic codes over Z_(p~(k+1)) and (1+p~k)- cyclic codes over Z_(p~(k+1)), the generator of (1+p~k)-cyclic codes are obtained.
In chapter 4, we focus on codes over Z_(p~2). First we note that the three class of codes-cyclic codes, constacyclic codes and quasi-cyclic codes, which are discussed in this thesis, may be nonlinear. In this chapter, the sufficient conditions for the Gray image of (1+p)- cyclic codes and the cyclic codes over Z_(p~2) to be linear axe also given.
本文继续对环Z_(p~(k+1))上的码展开研究，得到了以下主要结果．
在第二章，通过利用环Z_(p~(k+1))中的元素可以唯一写成p进制的形式，以及从Z_(p~(k+1))~n到Z_p~(p~kn)的Gray映射，我们给出了环Z_(p~(k+1))上的(1+p~k)-循环码的Gray像和一般循环码的Gray像以及负循环码的Gray像。
在第三章，我们给出了环Z_(p~(k+1))上长为n，(n，p)=1的常循环码的生成元．通过建立Z_(p~(k+1))上的循环码与(1+p~k)-循环码的一一对应给出了(1+p~k)-循环码的生成元．
在第四章，我们考虑环Z_(p~2)上的码．由于本文所说的循环码，常循环码，准循环码不一定是线性的．最后本文给出了环Z_(p~2)上(1+p)-循环码和循环码的Gray像是线性的充分条件．
Since 1970s, some researchers, such as Blake[1] and Speigel[2], began to discuss error-correcting codes over the rings Z_m of integers modulo m instead of studying codes over finite fields. In the beginning of 1990s, Forney etc[3] and Hammous etc[4] proved that some nonlinear codes, such as Kerdock code, Preparata code and Delsarte-Goethals code are the images of some linear codes over Z_4, where the map is the Gray map. Since these non-linear codes have more codewords than those linear codes with the same length and the same Hamming distance, the research interest in codes over finite rings has grown rapidly. In 1998, Carlet[6] defined Gray map over Z_(2~k) by using Boolean function, then mapped the linear codes over Z_(2~k) to nonlinear codes over Z_2 by the Gray map in this paper, and finally obtained generalized Kerdock code and generalized Goethals code. Ling[9] extended the definition of Gray map to Z_(p~(k+1)) and proved that the Gray image of (1-p~k)- cyclic codes are quasi-cyclic codes over Z_p. In [9], they also proved that cyclic codes over the rings Z_(p~(k+1)) are equivalent to quasi-cyclic codes by formulating a one-to-one correspondence between (1-p~k)-cyclic codes and general cyclic codes, they also provided a sufficient condition for the images of codes over Z_(p~(k+1)) to be linear.
In this thesis, we shall continue the study on codes over Z_(p~(k+1)). We obtain the following results.
In chapter 2, by using the Gray map (see [9]) from Z_(p~(k+1))~n to Z_p~(p~kn), and the unique p-adic expression of each element in Z_(p~(k+1)), the Gray images of (1+p~k)- cyclic codes of length n over the rings Z_(p~(k+1)), general cyclic codes and negacyclic codes are obtained, where (n,p) = 1.
In chapter 3, when (n,p) = 1, the generator of constacyclic codes with length n over Z_(p~(k+1)) are obtained. By using the one-to-one correspondence between cyclic codes over Z_(p~(k+1)) and (1+p~k)- cyclic codes over Z_(p~(k+1)), the generator of (1+p~k)-cyclic codes are obtained.
In chapter 4, we focus on codes over Z_(p~2). First we note that the three class of codes-cyclic codes, constacyclic codes and quasi-cyclic codes, which are discussed in this thesis, may be nonlinear. In this chapter, the sufficient conditions for the Gray image of (1+p)- cyclic codes and the cyclic codes over Z_(p~2) to be linear axe also given.
引文
[1]I. F. Blake, Codes over integer residue ring, Inform Contr., 29(1975), 295-300.
[2]E. Speigel, Codes over Z_m, Inform. Contr., 35(1977), 48-52.
[3]G. D. Forney, N. J. A. Sloane, M. D. Trott, The Nordstrom-Robinson code is a binaryImage of the Octacode, DIMACS, 14(1993), 19-26.
[4]A. R. Hammons, P. V. Kumar, A. R. Calderbank, et al, The Z4-linearity of Kerdock,Preparata, Goehals, and Related codes, IEEE Trans. Inform. Theory, 40(4) (1994),301-319.
[5]J. Wolfmann, Member IEEE. Negacyclic and cyclic codes over Z4, IEEE Trans. Inform.Theory, 45(7) (1999), 2527-2532.
[6]C. Carlet, Z_(2~k)-linear codes, IEEE Trans. Inform. Theory, 44(4)(1998), 1543-1547.
[7]H. Tapia-Recillas, G. Vega, Some constacyclic codes over Z_(2~k) and binary quasi-cycliccodes, Discrete Applied Math., 128(2003), 305-316.
[8]H. Tapia-Recillas, G. Vega, A generalization of negacyclic codes, in proc. int. workshopon coding and cryptography 2001, D. Augot and C. Carlet, Eds., 519-529.
[9]S. Ling, J. H. Blackford, Z_(p~(k+1))-linear codes, IEEE Tra.ns. Inform. Theory, 48(9) (2002),2592-2605.
[10]B. R. McDonald, Finite Rings with Identity, Marcel Dekker, inc., New York, 1974.
[11]P. Kanwar, R. Sergio, Cyclic codes over the integers modulo P~m, Finte Fields andTheir Appl, 3(1997), 334-352.
[12]冯倩倩，刘宏伟，环Z_(p~(k+1))上的常循环码，已接受，待发．
[13]J. Wolfmann, Binary images of cyclic codes over Z4, IEEE Trans. Inform. Theory, 47(5) (2001), 1773-1779.
[2]E. Speigel, Codes over Z_m, Inform. Contr., 35(1977), 48-52.
[3]G. D. Forney, N. J. A. Sloane, M. D. Trott, The Nordstrom-Robinson code is a binaryImage of the Octacode, DIMACS, 14(1993), 19-26.
[4]A. R. Hammons, P. V. Kumar, A. R. Calderbank, et al, The Z4-linearity of Kerdock,Preparata, Goehals, and Related codes, IEEE Trans. Inform. Theory, 40(4) (1994),301-319.
[5]J. Wolfmann, Member IEEE. Negacyclic and cyclic codes over Z4, IEEE Trans. Inform.Theory, 45(7) (1999), 2527-2532.
[6]C. Carlet, Z_(2~k)-linear codes, IEEE Trans. Inform. Theory, 44(4)(1998), 1543-1547.
[7]H. Tapia-Recillas, G. Vega, Some constacyclic codes over Z_(2~k) and binary quasi-cycliccodes, Discrete Applied Math., 128(2003), 305-316.
[8]H. Tapia-Recillas, G. Vega, A generalization of negacyclic codes, in proc. int. workshopon coding and cryptography 2001, D. Augot and C. Carlet, Eds., 519-529.
[9]S. Ling, J. H. Blackford, Z_(p~(k+1))-linear codes, IEEE Tra.ns. Inform. Theory, 48(9) (2002),2592-2605.
[10]B. R. McDonald, Finite Rings with Identity, Marcel Dekker, inc., New York, 1974.
[11]P. Kanwar, R. Sergio, Cyclic codes over the integers modulo P~m, Finte Fields andTheir Appl, 3(1997), 334-352.
[12]冯倩倩，刘宏伟，环Z_(p~(k+1))上的常循环码，已接受，待发．
[13]J. Wolfmann, Binary images of cyclic codes over Z4, IEEE Trans. Inform. Theory, 47(5) (2001), 1773-1779.