On linear complementary pairs of algebraic geometry codes over finite fields (Q6611739)

Let \(C\) be a linear code over the finite field \(\mathbb{F}_q\) of length \(n\) and \(C^{\perp}\) its dual code. The Euclidean hull of \(C\) is defined to be \(\mathrm{Hull}_E(C) := C \cap C^{\perp}\) and \(h_E(C)\) the dimension of \(\mathrm{Hull}_E(C)\). The code \(C\) is called an LCD code if \(h_E(C) = 0\). A pair of linear codes \((C,D)\) over \(\mathbb{F}_q\) of length \(n\) is called LCP if \(C\oplus D = \mathbb{F}_q^n\). When \(D = C^{\perp}\), then \(C\) is an LCD code. While significant results have been obtained for LCD codes, only partial results are known for LCP codes.\N\NIn this paper, some classes of LCP codes derived from algebraic curves are explicitly constructed. The security parameters of the derived LCP of codes \((C,D)\) which are determined by the minimum distances \(d(C)\) and \(d(D^{\perp})\) are studied. Furthermore, it is shown that for LCP algebraic geometry codes \((C,D)\), the dual code \(C^{\perp}\) is equivalent to \(D\) under some specific conditions, and the existence of MDS LCP of algebraic geometry codes is investigated. Moreover, the open problem of whether any pair of codes \((C, D)\) over a finite field \(\mathbb{F}_q\) with \(q > 2\) is equivalent to an LCP of codes is answered.