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# Duadic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$

• * Corresponding author: Maheshanand Bhaintwal
• In this paper, we study the structure of duadic codes of an odd length $n$ over $\mathbb{Z}_4+u\mathbb{Z}_4$, $u^2 = 0$, (more generally over $\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $u^2 = 0$, where $q = p^r$, $p$ a prime and $(n, p) = 1$) using the discrete Fourier transform approach. We study these codes by considering them as a class of abelian codes. Some results related to self-duality and self-orthogonality of duadic codes are presented. Some conditions on the existence of self-dual augmented and extended duadic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ are determined. We present a sufficient condition for abelian codes of the same length over $\mathbb{Z}_4+u\mathbb{Z}_4$ to have the same minimum Hamming distance. A new Gray map over $\mathbb{Z}_4+u\mathbb{Z}_4$ is defined, and it is shown that the Gray image of an abelian code over $\mathbb{Z}_4+u\mathbb{Z}_4$ is an abelian code over $\mathbb{Z}_4$. We have obtained five new linear codes of length $18$ over $\mathbb{Z}_4$ from duadic codes of length $9$ over $\mathbb{Z}_4+u\mathbb{Z}_4$ through the Gray map and a new map from $\mathbb{Z}_4+u\mathbb{Z}_4$ to $\mathbb{Z}_4^2$.

Mathematics Subject Classification: Primary: 94B05, 94B15.

 Citation:

• Table 1.  Duadic codes of $R(\mathbb{Z}_3\times \mathbb{Z}_3)$

 $\text{Duadic code}\; C$ $|C|$ $\psi(C)$ $\phi(C)$ $2-1-0-1-0$ $2^{18}$ $[18, 4^82^2, 4 ]^*$ $[18, 4^42^5, 8]^{**}$ $u-1-0-1-0$ $2^{18}$ $[18, 4^92^0, 4 ]$ $[18, 4^42^5, 6]$ $(2+u)-1-0-1-0$ $2^{18}$ $[18, 4^92^0, 4 ]$ $[18, 4^42^5, 6]$ $2-2-2-2-2$ $2^{18}$ $[18,4^02^{18},2 ]^*$ $[18, 4^02^9, 4]^\dagger$ $(2+u)-2-2-2-2$ $2^{18}$ $[18, 4^12^{16}, 4]^*$ $[18, 4^02^9, 8]^{\dagger **}$ $u-2-2-2-2$ $2^{18}$ $[18, 4^12^{16}, 4]$ $[18, 4^02^9, 8]^{\dagger}$ $2-2-u-u-2$ $2^{18}$ $[18, 4^42^{10},4 ]^{**}$ $[18, 4^02^9,8]$ $u-2-u-u-2$ $2^{18}$ $[18, 4^52^8, 4]^{**}$ $[18, 4^02^9,6]^\dagger$ $(2+u)-2-u-u-2$ $2^{18}$ $[18,4^5 2^8, 4 ]$ $[18, 4^02^9,8]$ $(2+u) - (2+u) -0-1- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^22^5, 6]^{**}$ $(2+u) - (2+u) -u-u- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^02^9,6]$ $(2+u) - (2+u) -2-2- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^02^9,6]$ $(2+u) - (2+u) -(2+u) - (2+u) - (2+u)$ $2^{18}$ $[18, 4^9 2^0, 2 ]$ $[18, 4^02^9,2]^\dagger$ $u - (2+u) -2-2- (2+u)$ $2^{18}$ $[18, 4^9 2^0, 4 ]$ $[18, 4^02^9,8]$
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