LCD codes from weighing matrices

Abstract

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.2010 Mathematics Subject Classification: 05B20, 05B30, 94B05, 12E20.

A new construction of quadratic double circulant LCD codes

“…Therefore, the linear codes with the generator matrix G = [Q q (1, 0, 1)|Q q (0, 1, 1)] or G = [Q q (1, 1, 0)|Q q (0, 1, 1)] are LCD codes over GF (l). The parameters of both codes are [14,7,4]. Theorem 3.15.…”

“…Meanwhile, Prakash et al [25] presented LCD codes over the ring F q + uF q and expounded an application of Hermitian LCD codes in the multi-secret sharing scheme, which was first presented for Euclidean LCD codes by Alahmadi et al [1]. Recently, LCD codes have been studied by using weighing matrices, and adjacency matrices in [7,8] using the concepts of (r, λ) design and strongly regular graphs (SRGs) or doubly regular tournament (DRTs), respectively. Most of the above works on double circulant LCD codes have been studied in terms of generator polynomials.…”