ACD codes over non-symmetric dualities
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The applications of additive codes mainly lie in quantum error correction and quantum computing. Due to their applications in quantum codes, additive codes have grown in importance. In addition to this, additive codes allow the implementation of a variety of dualities. The article begins by developing the properties of Additive Complementary Dual (ACD) codes with respect to arbitrary dualities over finite abelian groups. Further, we calculate precisely the total number of dualities over finite fields and introduce a new class of non-symmetric dualities, denoted as class A. Two conditions have been obtained, one is necessary and sufficient condition and other is a necessary condition. The necessary and sufficient condition is for an additive code to be an ACD code over arbitrary dualities, along with an algorithm for determining whether an additive code is an ACD code or not. The necessary condition is on the generator matrix of an ACD code for any duality belonging to the class A. We provide bounds for the highest possible distance of ACD codes over finite fields. Finally, we examine non-symmetric dualities over F4.
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