• Title/Summary/Keyword: cyclic code over $\mathbb{Z}_4$

Search Result 5, Processing Time 0.019 seconds

CYCLIC CODES OF EVEN LENGTH OVER Z4

  • Woo, Sung-Sik
    • Journal of the Korean Mathematical Society
    • /
    • v.44 no.3
    • /
    • pp.697-706
    • /
    • 2007
  • In [8], we showed that any ideal of $\mathbb{Z}_4[X]/(X^{2^n}-1)$ is generated by at most two polynomials of the standard forms. The purpose of this paper is to find a description of the cyclic codes of even length over $\mathbb{Z}_4$ namely the ideals of $\mathbb{Z}_4[X]/(X^l\;-\;1)$, where $l$ is an even integer.

CYCLIC CODES OF LENGTH 2n OVER ℤ4

  • Woo, Sung Sik
    • Communications of the Korean Mathematical Society
    • /
    • v.28 no.1
    • /
    • pp.39-54
    • /
    • 2013
  • The purpose of this paper is to find a description of the cyclic codes of length $2^n$ over $\mathbb{Z}_4$. We show that any ideal of $\mathbb{Z}_4$[X]/($X^{2n}$ - 1) is generated by at most two polynomials of the standard forms. We also find an explicit description of their duals in terms of the generators.

FREE CYCLIC CODES OVER FINITE LOCAL RINGS

  • Woo, Sung-Sik
    • Bulletin of the Korean Mathematical Society
    • /
    • v.43 no.4
    • /
    • pp.723-735
    • /
    • 2006
  • In [2] it was shown that a 1-generator quasi-cyclic code C of length n = ml of index l over $\mathbb{Z}_4$ is free if C is generated by a polynomial which divides $X^m-1$. In this paper, we prove that a necessary and sufficient condition for a cyclic code over $\mathbb{Z}_pk$ of length m to be free is that it is generated by a polynomial which divides $X^m-1$. We also show that this can be extended to finite local rings with a principal maximal ideal.

IDEALS OF Zpn[X]/(Xl-1)

  • Woo, Sung-Sik
    • Communications of the Korean Mathematical Society
    • /
    • v.26 no.3
    • /
    • pp.427-443
    • /
    • 2011
  • In [6, 8], we showed that any ideal of $\mathbb{Z}_4[X]/(X^l\;-\;1)$ is generated by at most two polynomials of the `standard' forms when l is even. The purpose of this paper is to find the `standard' generators of the cyclic codes over $\mathbb{Z}_{p^a}$ of length a multiple of p, namely the ideals of $\mathbb{Z}_{p^a}[X]/(X^l\;-\;1)$ with an integer l which is a multiple of p. We also find an explicit description of their duals in terms of the generators when a = 2.

ALGEBRAS WITH A NILPOTENT GENERATOR OVER ℤp2

  • Woo, Sung-Sik
    • Bulletin of the Korean Mathematical Society
    • /
    • v.43 no.3
    • /
    • pp.487-497
    • /
    • 2006
  • The purpose of this paper is to describe the structure of the rings $\mathbb{Z}_{p^2}[X]/({\alpha}(X))$ with ${\alpha}(X)$ a monic polynomial and $\={X}^{\kappa}=0$ for some nonnegative integer ${\kappa}$. Especially we will see that any ideal of such rings can be generated by at most two elements of the special form and we will find the 'minimal' set of generators of the ideals. We indicate how to identify the isomorphism types of the ideals as $\mathbb{Z}_{p^2}-modules$ by finding the isomorphism types of the ideals of some particular ring. Also we will find the annihilators of the ideals by finding the most 'economical' way of annihilating the generators of the ideal.