@article{Aydogdu2024, 
author = {Ismail Aydogdu},
title = {On double cyclic codes over              Z        2    +  u            Z        2},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {5},
pages = {11076-11091},
keywords = {double cyclic codes, Z2 + uZ2, self-dual cyclic codes, optimal codes},
url = {https://www.sciopen.com/article/10.3934/math.2024543},
doi = {10.3934/math.2024543},
abstract = {In this paper, we introduced double cyclic codes over        R    r    ×      R    s  , where    R  =            Z              2        +  u            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    ×      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.}
}