A Class of Optimal Cyclic Codes With Two Zeros

Let <inline-formula> <tex-math notation="LaTeX">$m>2$ </tex-math></inline-formula> be an integer and <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> be an odd prime. We explore the minimum distance of <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>-ary cyclic codes of length <inline-formula> <tex-math notation="LaTeX">$n = 2(p^{m}-1)/(p-1)$ </tex-math></inline-formula> with two zeros. A sufficient condition for such cyclic codes with minimum distance at least three is obtained. A class of optimal <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>-ary cyclic codes with minimum distance four are presented. Four explicit constructions for such optimal cyclic codes are provided. The weight distribution of the dual of the cyclic code in the first construction is given.

[1]  Tor Helleseth,et al.  On a conjecture about a class of optimal ternary cyclic codes , 2015, 2015 Seventh International Workshop on Signal Design and its Applications in Communications (IWSDA).

[2]  Lisha Wang,et al.  Several classes of optimal p-ary cyclic codes with minimal distance four , 2022, ArXiv.

[3]  Jiejing Wen,et al.  A q-polynomial approach to constacyclic codes , 2017, Finite Fields Their Appl..

[4]  Xiwang Cao,et al.  Optimal p-ary cyclic codes with minimum distance four from monomials , 2016, Cryptography and Communications.

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

[6]  Cunsheng Ding,et al.  A class of three-weight cyclic codes , 2013, Finite Fields Their Appl..

[7]  Cunsheng Ding,et al.  Optimal ternary cyclic codes with minimum distance four and five , 2013, Finite Fields Their Appl..

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

[9]  Xiaoni Du,et al.  A family of optimal ternary cyclic codes from the Niho-type exponent , 2018, Finite Fields Their Appl..

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

[11]  Cunsheng Ding,et al.  Optimal Ternary Cyclic Codes From Monomials , 2013, IEEE Transactions on Information Theory.

[12]  Jin Li,et al.  On the minimum distance of negacyclic codes with two zeros , 2019, Finite Fields Their Appl..

[13]  Victor Zinoviev,et al.  On the Minimum Distances of Non-Binary Cyclic Codes , 1999, Des. Codes Cryptogr..

[14]  Tor Helleseth,et al.  New Families of Almost Perfect Nonlinear Power Mappings , 1999, IEEE Trans. Inf. Theory.

[15]  Zhengchun Zhou,et al.  A class of optimal ternary cyclic codes and their duals , 2015, Finite Fields Their Appl..

[16]  Victor Zinoviev,et al.  ON BINARY CYCLIC CODES WITH MINIMUM DISTANCE D = 3 , 1997 .

[17]  Cunsheng Ding,et al.  A q-polynomial approach to cyclic codes , 2013, Finite Fields Their Appl..

[18]  Cunsheng Ding,et al.  LCD Cyclic Codes Over Finite Fields , 2017, IEEE Transactions on Information Theory.