Reed-Muller-type codes over the Segre variety
The image of the Segre embedding \(\mathbb P^n_k \times \mathbb P^m_k\rightarrow \mathbb P^N_k\), where \(N=(n+1)(m+1)-1\), is the Segre variety \(\Sigma_{n,m}\). If \(K\) is a finite field, the authors consider the evaluation code \(C(d)\) obtained by evaluating homogeneous polynomials of degree \(d\) on (representatives of) the \(K\)-rational points of \(\Sigma_{n,m}\). The Hilbert function of \(\Sigma_{n,m}\) is used to compute the dimension of \(C(d)\). The authors show that the minimum distance of \(C(d)\) is \(\delta_{\mathbb P^n}(d)\delta_{\mathbb P^m}(d)\), where \(\delta_{\mathbb P^l}(d)\) is the minimum distance of the projective Reed-Muller code obtained by evaluating homogeneous polynomials of degree \(d\) on the \(K\)-rational points of \(\mathbb P^l\) (cf. \textit{G. Lachaud} [Projective Reed-Muller codes. In: Coding theory and applications, Proc. 2nd Int. Colloq., Paris/Fr. 1986, Lect. Notes Comput. Sci. 311, 125--129 (1988; Zbl 0651.94008)] and \textit{A. B. Sørensen} [IEEE Trans. Inf. Theory 37, 1567--1576 (1991; Zbl 0741.94016)]). An example is given with \(n=m=1\), \(d=2\) and \(K={\mathbb F}_4\).
scientific article; zbMATH DE number 1891463
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