LCD codes from weighing matrices
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Linear codes with complementary duals are linear codes whose intersection with their duals are trivial, shortly named LCD codes. In this paper we outline a construction for LCD codes over finite fields of order q using weighing matrices and their orbit matrices. The LCD codes constructed can be of any length dimension according to the choice of matrices used in their construction. As a special case, LCD codes of length 2n and dimension n are constructed which also have the property of being formally self-dual. Alternatively, under a condition depending on q that the codes are not LCD, this method constructs self-dual codes. To illustrate the method we construct LCD codes from weighing matrices, including the Paley conference matrices and Hadamard matrices. We also extend the construction to Hermitian LCD codes over the finite field of order 4. In addition, we propose a decoding algorithm that can be feasible for the LCD codes obtained from some of the given methods.
KeywordsWeighing matrix Orbit matrix LCD code
Mathematics Subject Classification05B20 05B30 94B05 12E20
D. Crnković and A. Švob were supported by Croatian Science Foundation under the project 6732. R. Egan was supported by the Irish Research Council (Government of Ireland Postdoctoral Fellowship, GOIPD/2018/304). B. G. Rodrigues work is based on the research supported by the National Research Foundation of South Africa (Grant Nos. 95725 and 106071). B. G. Rodrigues acknowledges support from the Erasmus Mundus Plus academic exchange programme to visit the University of Rijeka in 2018.
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