In this paper, we introduced double cyclic codes over $ R^r\times R^s $, where $ R = \mathbb{Z}_{2}+u\mathbb{Z}_{2} = \{0, 1, u, 1+u\} $ is the ring with four elements and $ u^2 = 0 $. We first determined the generator polynomials of $ R $-double cyclic codes for odd integers $ r $ and $ s $, then gave the generators of duals of free double cyclic codes over $ R^r\times R^s $. By defining a linear Gray map, we looked at the binary images of $ R $-double cyclic codes and gave several examples of optimal parameter binary linear codes obtained from $ R $-double cyclic codes. Moreover, we studied self-dual $ R $-double cyclic codes and presented an example of a self-dual $ R $-double cyclic code.
Citation: Ismail Aydogdu. On double cyclic codes over $ \mathbb{Z}_2+u\mathbb{Z}_2 $[J]. AIMS Mathematics, 2024, 9(5): 11076-11091. doi: 10.3934/math.2024543
In this paper, we introduced double cyclic codes over $ R^r\times R^s $, where $ R = \mathbb{Z}_{2}+u\mathbb{Z}_{2} = \{0, 1, u, 1+u\} $ is the ring with four elements and $ u^2 = 0 $. We first determined the generator polynomials of $ R $-double cyclic codes for odd integers $ r $ and $ s $, then gave the generators of duals of free double cyclic codes over $ R^r\times R^s $. By defining a linear Gray map, we looked at the binary images of $ R $-double cyclic codes and gave several examples of optimal parameter binary linear codes obtained from $ R $-double cyclic codes. Moreover, we studied self-dual $ R $-double cyclic codes and presented an example of a self-dual $ R $-double cyclic code.
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Ismail Aydogdu. On double cyclic codes over $ \mathbb{Z}_2+u\mathbb{Z}_2 $[J]. AIMS Mathematics, 2024, 9(5): 11076-11091. doi: 10.3934/math.2024543
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