LCD codes from weighing matrices

• Published in: Applicable algebra in engineering, communication and computing Vol. 32; no. 2; pp. 175 - 189
• Main Authors: Crnković, Dean, Egan, Ronan, Rodrigues, B. G., Švob, Andrea
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2021; Springer Nature B.V
• Subjects: Algorithms; Artificial Intelligence; Binary system; Codes; Computer Hardware; Computer Science; Decoding; Fields (mathematics); Linear codes; Mathematical analysis; Matrix methods; Original Paper; Symbolic and Algebraic Manipulation; Theory of Computation; Weighing; Orbit matrix; LCD code; 05B20; 05B30; 94B05; 12E20; Weighing matrix
• ISSN: 0938-1279; 1432-0622
• DOI: 10.1007/s00200-019-00409-8

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 2 n 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.