Distinct coordinate solutions of linear equations over finite fields

Abstract Let F q be the finite field of q elements and a 1 , a 2 , … , a k , b ∈ F q . We investigate N F q ( a 1 , a 2 , … , a k ; b ) , the number of ordered solutions ( x 1 , x 2 , … , x k ) ∈ F q k of the linear equation a 1 x 1 + a 2 x 2 + ⋯ + a k x k = b with all x i distinct. We obtain an explicit formula for N F q ( a 1 , a 2 , … , a k ; b ) involving combinatorial numbers depending on a i 's. In particular, we obtain closed formulas for two special cases. One is that a i , 1 ≤ i ≤ k take at most three distinct values and the other is that ∑ i = 1 k a i = 0 and ∑ i ∈ I a i ≠ 0 for any I ⊊ { 1 , 2 , … , k } . The same technique works when F q is replaced by Z n , the ring of integers modulo n. In particular, we give a new proof for the main result given by Bibak, Kapron and Srinivasan ( [2] ), which generalizes a theorem of Schonemann via a graph theoretic method.