Self-dual additive codes
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Self‑dual additive codes Steven T. Dougherty1 · Adrian Korban2 · Serap Şahinkaya3 Received: 12 March 2020 / Accepted: 4 November 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group ℤpe . They exist for all lengths when p is prime and e is even; all even lengths when p is an odd prime with p ≡ 1 (mod 4) and e is odd with e > 1 ; and all lengths that are 0 (mod 4) when p is an odd prime with p ≡ 3 (mod 4) and e is odd with e > 1. Keywords Self-dual codes · Finite abelian groups · Group characters MSC 11T71 94B05
1 Introduction Self-dual codes are one of the most widely studied topics in algebraic coding theory. They have been shown to have numerous applications to lattice theory, design theory, and invariant theory, as well as being of great interest as mathematical objects in themselves. Classically, self-dual codes have been defined with respect to the standard Euclidean inner-product or the Hermitian inner-product over finite fields. Later this was generalized to codes that were submodules of Rn , where R is a finite Frobenius ring, see [5] for a complete description in the commutative case and [8] for a complete description in the non-commutative case. Generally, this was the broadest application in this setting for self-dual codes, since for non-Frobenius rings * Steven T. Dougherty [email protected] 1
Department of Mathematics, University of Scranton, Scranton, PA 18518, USA
2
Department of Mathematical and Physical Sciences, University of Chester, Thornton Science Park, Pool Ln, Chester CH2 4NU, England
3
Technology Faculty, Department of Mathematics and Natural Sciences, Tarsus University, Mersin, Turkey
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we lose the usual cardinality condition for a code and its orthogonal, namely that the product of the cardinality of the code and its dual must be the cardinality of the ambient space. In these works, it was shown precisely under which conditions selfdual codes exist. In this work, we extend this study to codes over additive groups with any duality. Thus we generalize both the ambient space and the type of innerproducts allowed. Namely, we give results with respect to any duality which means that the results hold for any inner-product satisfying certain canonical requirements. Additive codes have become an increasingly important topic in coding theory since they have applications in quantum error-correction and quantum computing, see [3] for foundational results in this vein and [1, 2, 11] and [12] for examples. They are codes that are defined over an additive group. When this group is the additive group of a finite field or a ring, additive codes need not be linear,
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