On Constacyclic Codes over $\boldsymbol{Z}_{\boldsymbol{p}_{1}\boldsymbol{p}_{2}\cdots\boldsymbol{p}_{\boldsymbol{t}}}$

Abstract

Let t ≥ 2 be an integer, and let p ₁, ⋯, p _t be distinct primes. By using algebraic properties, the present paper gives a sufficient and necessary condition for the existence of non-trivial self-orthogonal cyclic codes over the ring ${Z_{{p_1}{p_2} \cdots {p_t}}}$ and the corresponding explicit enumerating formula. And it proves that there does not exist any self-dual cyclic code over ${Z_{{p_1}{p_2} \cdots {p_t}}}$.

Keywords

Ideal / Isomorphism / Constacyclic code / Self-orthogonal code / Self-orthogonal cyclic code

References

[1] Berlekamp E R. Negacyclic Codes for the Lee Metric, 1967, Chapel Hill, N.C.: North Carolina Press

[2] Blackford T. Negacyclic codes over Z₄ of even length. IEEE Trans. Inform. Theory, 2003, 49(6): 1417-1424

[3] Cao Y L. On constacyclic codes over finite chain rings. Finite Fields and Their Applications, 2013, 24: 124-135

[4] Dinh H Q. Constacyclic codes of length p ^s over ${F_{{p^m}}} + u{F_{{p^m}}}$. Journal of Algebra, 2010, 324(5): 940-950

[5] Dinh H Q, Wang L Q, Zhu S X. Negacyclic codes of length 2p ^s over ${F_{{p^m}}} + u{F_{{p^m}}}$. Finite Fields and Their Applications, 2015, 31: 178-201

[6] Garrett P. The Mathematics of Coding Theory, 2005, Beijing: China Machine Press

[7] Hammons R, Kumar P V, Calderbank A R The Z ₄-linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inform. Theory, 1994, 40(2): 301-319

[8] Huffman W C, Pless V. Fundamentals of Error-Correcting Codes, 2003, New York: Cambridge University Press

[9] Jacobson N. Basic Algebra I, 1985, New York: W. H. Freeman and Company

[10] Jia Y, Ling S, Xing C P. On self-dual cyclic codes over finite fields. IEEE Trans. Inform. Theory, 2011, 57(4): 2243-2251

[11] Liao Q Y, Li Y B, Liao H. The existence for self-orthogonal cyclic codes over finite fields. Acta Mathematica Sinica, 2014, 57(1): 117-124

[12] Nechaev A A. Kerdock code in a cyclic form. Discrete Mathematics and Applications, 1991, 1(4): 365-384

[13] Pan C D, Pan C B. Elementary Number Theory, 1992, Beijing: Peking University Press

[14] Prange E. Cyclic Error-Correcting Codes in Two Symbols, 1957, Cambridge: Air Force Cambridge Research Center

[15] Zhu S X, Kai X S. Dual and self-dual negacyclic codes of even length over ${Z_{{2^a}}}$. Discrete Mathematics, 2009, 309(8): 2382-2391