Cayley-Bacharach and evaluation codes on complete intersections (Q1763646)

Let \(\mathbb F_q\) be the finite field of order \(q\) and take \(\Gamma=\{p_1,p_2,\ldots,p_n\}\) as a finite set of cardinality \(n\). Suppose that we are given a \(\mathbb F_q\)-vector space \(R\) of functions \(X\rightarrow \mathbb F_q\). Then, the set \[ C=\{ (f(p_1),f(p_2),\ldots,f(p_n)): f\in R \} \] is a linear code of length \(n\). Let now \(f_0\) be a polynomial of degree \(a\) in \(\mathbb F_q [x_0,x_1,\ldots,x_m]\) such that \(f_0(p_i)\neq 0\) for all \(1\leq i\leq n\). This paper is concerned with the case in which \(\Gamma\) is the reduced complete intersection of \(m\) hypersurfaces of \(\mathbb P^m\) with degrees respectively \(d_1, d_2, \ldots, d_m\) and \[ R=R_a=\left\{\left({{f(p_1)}\over{f_0(p_1)}}, {{f(p_2)}\over{f_0(p_2)}},\ldots,{{f(p_n)}\over{f_0(p_n)}} \right) : f\in \mathbb F_q [x_0,x_1,\ldots,x_m], \deg(f)=a\right\}. \] The main theorem, obtained using the Cayley-Bacharach theorem, is a lower bound of the minimum distance on the code \(C_a\) thus obtained, namely that if \[ 1\leq a\leq s=\sum_{i=1}^m d_i-m-1, \] then the minimum distance \(d\) of \(C_a\) satisfies \(d\geq s-a+2\). Several examples of application of this theorem are then provided.