International Journal of Engineering Research and Development ISSN: 2278-067X, Volume 1, Issue 9 (June 2012), PP.34-37 www.ijerd.com 34 Cyclic Codes of Length 2 k Over Z 2 m Arpana Garg 1 , Sucheta Dutt 2 1,2 Department of Applied Science, PEC University of Technology, Sector – 12, Chandigarh, India. Abstract—In this paper, the structure of cyclic codes over m Z 2 of length k 2 for any natural number k is studied. It is proved that cyclic codes over     1 2 n x x Z R m / ] [ of length n = k 2 are generated by at most m elements. Keywords— Codes, Cyclic codes, Ideals, Principal ideal Ring I. INTRODUCTION Let R be a commutative finite ring with identity. A linear code C over R of length n is defined as a R-submodule of n R . A cyclic code C over R of length n is a linear code such that any cyclic shift of a codeword is also a codeword, that is, whenever ) ,..., , , ( n c c c c 3 2 1 is in C then so is ) ,..., , , ( 1 2 1  n n c c c c . Most of the work on cyclic codes over 4 Z has been done in [2,6,7]. Cyclic codes over ring m Z are studied by Abualrub in [3] where length of code is relatively prime to m. Structure of Cyclic codes over 2 p Z of length e p is studied in [8,12]. T.Abualrub and I.Siap give the structure of cyclic codes over rings of characteristic 2, that is, 2 2 uZ Z  and 2 2 2 2 Z u uZ Z   in [11]. In [10], Structure of cyclic codes over 8 Z is given. A class of constacyclic codes is studied by Dinh in [1] and Zhu in [13]. In this paper, we study the structure of cyclic codes of length k 2 over m Z 2 and prove that cyclic codes of length k 2 over m Z 2 are generated by at most m elements. II. PRELIMINARIES Codewords of a cyclic code of length n over a ring R can be represented by polynomials modulo 1  n x . Thus any codeword ) ,..., , , ( 1 2 1 0   n c c c c c can be represented by polynomial 1 1 2 2 1 0        n n x c x c x c c x c ... ) ( in the ring R. 2.1 Definition : Define a map       1 1 2 2 n n x x Z x x Z m / ] [ / ] [ : φ such that φ maps zero divisors in m Z 2 to 0; and units of m Z 2 to 1; and x to x. It is easy to prove that φ is an epimorphism of rings. Any polynomial     1 2 n x x Z x f m / ] [ ) ( can be represented as ) ( ... ) ( ) ( ) ( ) ( x f x f x f x f x f m m 1 3 2 2 1 2 2 2       where i x x Z x f n i      1 2 / ] [ ) ( . The image of ) ( x f under φ is ) ( x f 1 . 2.2. Definition [9]: The content of the polynomial m m x a x a x a a x f      ... ) ( 2 2 1 0 where the a i ’s belong to m Z 2 , is the greatest common divisor of a 0 ,a 1 ,a 2 ,...,a m . 2.3. Lemma [2]: If R is a local ring with the unique maximal ideal M and M=(a 1 ,a 2 ,...,a n )=<a>, then M=<a i > for some i. Consider the ring     1 2 n x x Z R m / ] [ . Let C be an ideal in R of length k 2 over m Z 2 . The following lemmas can be easily proved. 2.4. Lemma: R is a local ring with the unique maximal ideal    1 2 x M , . 2.5. Lemma: R is not a Principal ideal ring.