A sequence construction of cyclic codes over finite fields

Due to their efficient encoding and decoding algorithms, cyclic codes, a subclass of linear codes, have applications in communication systems, consumer electronics, and data storage systems. There are several approaches to constructing all cyclic codes over finite fields, including the generator matrix approach, the generator polynomial approach, and the generating idempotent approach. Another one is a sequence approach, which has been intensively investigated in the past decade. The objective of this paper is to survey the progress in this direction in the past decade. Many open problems are also presented in this paper.

[1]  Cunsheng Ding,et al.  Linear codes from perfect nonlinear mappings and their secret sharing schemes , 2005, IEEE Transactions on Information Theory.

[2]  G. David Forney,et al.  On decoding BCH codes , 1965, IEEE Trans. Inf. Theory.

[3]  Cunsheng Ding,et al.  Cyclic Codes from Some Monomials and Trinomials , 2013, SIAM J. Discret. Math..

[4]  Susanne Ebersbach,et al.  Designs And Their Codes , 2016 .

[5]  Cunsheng Ding,et al.  Cyclic codes from cyclotomic sequences of order four , 2012, Finite Fields Their Appl..

[6]  Leopold Bömer,et al.  Complex sequences over GF(pM) with a two-level autocorrelation function and a large linear span , 1992, IEEE Trans. Inf. Theory.

[7]  Carlos R. P. Hartmann,et al.  Generalizations of the BCH Bound , 1972, Inf. Control..

[8]  L. Dickson.  The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group. , 1896 .

[9]  Cunsheng Ding,et al.  Cyclic Codes From the Two-Prime Sequences , 2012, IEEE Transactions on Information Theory.

[10]  Maozhi Xu,et al.  A note on cyclic codes from APN functions , 2014, Applicable Algebra in Engineering, Communication and Computing.

[11]  Joseph L. Yucas,et al.  Dickson polynomials , 2013, Handbook of Finite Fields.

[12]  C. Ding.  Codes From Difference Sets , 2014 .

[13]  Cunsheng Ding,et al.  A family of skew Hadamard difference sets , 2006, J. Comb. Theory, Ser. A.

[14]  Cunsheng Ding,et al.  Binary cyclic codes from explicit polynomials over GF(2m) , 2013, Discret. Math..

[15]  E. Weldon.  Difference-set cyclic codes , 1966 .

[16]  Robert S. Coulter,et al.  Planar Functions and Planes of Lenz-Barlotti Class II , 1997, Des. Codes Cryptogr..

[17]  Cunsheng Ding,et al.  The weight distribution of a class of linear codes from perfect nonlinear functions , 2006, IEEE Transactions on Information Theory.

[18]  Cunsheng Ding.  Cyclic Codes from Dickson Polynomials , 2012, ArXiv.

[19]  Cunsheng Ding,et al.  The dimension and minimum distance of two classes of primitive BCH codes , 2016, Finite Fields Their Appl..

[20]  Pascale Charpin,et al.  Open problems on cyclic codes , 2009 .

[21]  Cunsheng Ding,et al.  The Stability Theory of Stream Ciphers , 1991, Lecture Notes in Computer Science.

[22]  Robert T. Chien,et al.  Cyclic decoding procedures for Bose- Chaudhuri-Hocquenghem codes , 1964, IEEE Trans. Inf. Theory.

[23]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .