Linearized Reed-Solomon codes and linearized Wenger graphs

Let m, d and k be positive integers such that k ≤ me , where e = (m, d). Let p be an prime number and π a primitive element of Fpm . To each ~a = (a0, · · · , ak−1) ∈ F k pm , we associate the linearized polynomial f~a(x) = k−1 ∑ j=0 ajx p . And, to each f~a(x), we associated the sequence c~a = (f~a(1), f~a(π), · · · , f~a(π pm−2)). Let C = {c~a | ~a ∈ F k pm} be the cyclic code formed by the sequences c~a’s. We call the dual code of C a linearized Reed-Solomon code. The weight distribution of the code C is determined in the present paper. Associated to the k-tuple g = (xy, x d y, · · · , x (k−1)d y) of polynomials in Fpm [x, y], there is a Wenger graph Wpm(g). The spectrum of the graph Wpm(g) is also determined in the present paper.