Weight distributions of a class of cyclic codes with arbitrary number of nonzeros in quadratic case

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. So far, most of previous results obtained were for cyclic codes with no more than three nonzeros. Recently, the authors of 37 constructed a class of cyclic codes with arbitrary number of nonzeros, and computed the weight distribution for several cases. In this paper, we determine the weight distribution for a new family of such codes. This is achieved by introducing certain new methods, such as the theory of Jacobi sums over finite fields and subtle treatment of some complicated combinatorial identities.

[1]  Gerardo Vega,et al.  The Weight Distribution of an Extended Class of Reducible Cyclic Codes , 2012, IEEE Transactions on Information Theory.

[2]  Cunsheng Ding,et al.  The Weight Enumerator of a Class of Cyclic Codes , 2011, IEEE Transactions on Information Theory.

[3]  Jing Yang,et al.  Weight Distribution of a Class of Cyclic Codes With Arbitrary Number of Zeros , 2013, IEEE Transactions on Information Theory.

[4]  Lei Hu,et al.  Weight Distribution of A p-ary Cyclic Code , 2009, ArXiv.

[5]  Gennian Ge,et al.  The Weight Distribution of a Class of Cyclic Codes Related to Hermitian Forms Graphs , 2012, IEEE Transactions on Information Theory.

[6]  Bernhard Schmidt,et al.  All Two-Weight Irreducible Cyclic Codes? , 2002 .

[7]  R. McEliece.  Irreducible Cyclic Codes and Gauss Sums , 1975 .

[8]  Robert J. McEliece,et al.  Weights of Irreducible Cyclic Codes , 1972, Inf. Control..

[9]  Keqin Feng,et al.  On the Weight Distributions of Two Classes of Cyclic Codes , 2008, IEEE Transactions on Information Theory.

[10]  Philippe Langevin,et al.  On the Weights of Binary Irreducible Cyclic Codes , 2005, WCC.

[11]  Robert W. Fitzgerald,et al.  Sums of Gauss sums and weights of irreducible codes , 2005, Finite Fields Their Appl..

[12]  Jacques Wolfmann,et al.  Weight distributions of some binary primitive cyclic codes , 1994, IEEE Trans. Inf. Theory.

[13]  Tao Feng.  On cyclic codes of length $${2^{2^r}-1}$$ with two zeros whose dual codes have three weights , 2012, Des. Codes Cryptogr..

[14]  H. Hollmann,et al.  On Binary Cyclic Codes With Few Weights , 2001 .

[15]  Cunsheng Ding,et al.  The Weight Distribution of Some Irreducible Cyclic Codes , 2009, IEEE Transactions on Information Theory.

[16]  Philippe Delsarte,et al.  On subfield subcodes of modified Reed-Solomon codes (Corresp.) , 1975, IEEE Trans. Inf. Theory.

[17]  Jing Yang,et al.  Weight Distributions of a Class of Cyclic Codes with Arbitrary Number of Zeros II , 2014, ArXiv.

[18]  Keqin Feng,et al.  Cyclic Codes and Sequences From Generalized Coulter–Matthews Function , 2008, IEEE Transactions on Information Theory.

[19]  Cunsheng Ding,et al.  The Weight Distributions of the Duals of Cyclic Codes With Two Zeros , 2011, IEEE Transactions on Information Theory.

[20]  Asha Rao,et al.  A Family of Two-Weight Irreducible Cyclic Codes , 2010, IEEE Transactions on Information Theory.

[21]  Yuansheng Tang,et al.  Cyclic Codes and Sequences: The Generalized Kasami Case , 2009, IEEE Transactions on Information Theory.

[22]  Maosheng Xiong,et al.  The weight distributions of a class of cyclic codes II , 2011, Finite Fields Their Appl..

[23]  Keqin Feng,et al.  Weight distribution of some reducible cyclic codes , 2008, Finite Fields Their Appl..

[24]  R. Schoof.  Families of curves and weight distributions of codes , 1995, math/9504222.

[25]  G. Myerson.  Period polynomials and Gauss sums for finite fields , 1981 .

[26]  M. V. Vlugt.  Hasse-Davenport Curves, Gauss Sums, and Weight Distributions of Irreducible Cyclic Codes , 1995 .

[27]  Tao Feng,et al.  Evaluation of the Weight Distribution of a Class of Cyclic Codes Based on Index 2 Gauss Sums , 2012, IEEE Transactions on Information Theory.

[28]  Lei Hu,et al.  The weight distribution of a class of p-ary cyclic codes , 2010, Finite Fields Their Appl..

[29]  Marko J. Moisio,et al.  Explicit evaluation of some exponential sums , 2009, Finite Fields Their Appl..

[30]  S. Lang.  Number Theory III , 1991 .

[31]  Robert Segal,et al.  Weight distributions of some irreducible cyclic codes , 1986 .

[32]  Cunsheng Ding,et al.  Hamming weights in irreducible cyclic codes , 2011, Discret. Math..

[33]  R. Calderbank,et al.  The Geometry of Two‐Weight Codes , 1986 .

[34]  Gerardo Vega,et al.  Determining the Number of One-Weight Cyclic Codes When Length and Dimension Are Given , 2007, WAIFI.

[35]  Yixian Yang,et al.  The Weight Distributions of Cyclic Codes and Elliptic Curves , 2011, IEEE Transactions on Information Theory.

[36]  Torleiv Kløve,et al.  Codes for Error Detection , 2007, Series on Coding Theory and Cryptology.

[37]  Maosheng Xiong.  The weight distributions of a class of cyclic codes II , 2014, Des. Codes Cryptogr..

[38]  M. Moisio,et al.  On the weight distribution of the duals of irreducible cyclic codes , cyclic codes with two zeros and hyper-Kloosterman codes , 2007 .

[39]  Maosheng Xiong,et al.  The weight distributions of a class of cyclic codes III , 2012, Finite Fields Their Appl..

[40]  Robert P. Langlands,et al.  Euler Products , 2022 .

[41]  K. Williams,et al.  Gauss and Jacobi sums , 2021, Mathematical Surveys and Monographs.

[42]  Gerardo Vega,et al.  New classes of 2-weight cyclic codes , 2007, Des. Codes Cryptogr..

[43]  R. McEliece,et al.  Euler products, cyclotomy, and coding☆ , 1972 .