Abstract
In this article, we show explicitly all possible weight enumerators for every irreducible cyclic code of length \(n\) over a finite field \({\mathbb {F}}_q\), in the case which each prime divisor of \(n\) is also a divisor of \(q-1\).
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Brochero Martínez F.E., Giraldo Vergara C.R., Batista de Oliveira L.: Explicit factorization of \(x^{n}-1\in {\mathbb{F}}_q[x]\). Des. Codes Cryptogr. (Accepted).
Chen B., Li L., Tuerhong R.: Explicit factorization of \(x^{2^mp^n}-1 \) over a finite field. Finite Fields Appl. 24, 95–104 (2013).
Ding C.: The weight distribution of some irreducible cyclic codes. IEEE Trans. Inf. Theory 55, 955–960 (2009).
Ding C., Yang J.: Hamming weights in irreducible cyclic codes. Discret. Math. 313, 434–446 (2013).
Farrell P.G., Castieira Moreira J.: Essentials of Error-Control Coding. Wiley, Hoboken (2006).
Fitzgerald R.W., Yucas J.L.: Explicit factorization of cyclotomic and Dickson polynomials over finite fields. In Arithmetic of Finite Fields. Lecture Notes in Computer Science, vol. 4547, pp. 1–10. Springer, Berlin (2007).
Golay M.J.E.: Notes on digital coding. Proc. IRE 37, 657 (1949).
Lidl R., Niederreiter H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20. Addison-Wesley, Boston (1983).
Meyn H.: Factorization of the cyclotomic polynomials \(x^{2^n}+1\) over finite fields. Finite Fields Appl. 2, 439–442 (1996).
Sharma A., Bakshi G.: The weight distribution of some irreducible cyclic codes. Finite Fields Appl. 18, 144–159 (2012).
Wang L., Wang Q.: On explicit factors of cyclotomic polynomials over finite fields. Des. Codes Cryptogr. 63(1), 87–104 (2012).
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Communicated by J. D. Key.
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Brochero Martínez, F.E., Giraldo Vergara, C.R. Weight enumerator of some irreducible cyclic codes. Des. Codes Cryptogr. 78, 703–712 (2016). https://doi.org/10.1007/s10623-014-0026-6
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DOI: https://doi.org/10.1007/s10623-014-0026-6