Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three
A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of Z(n) such that Delta(x) boolean AND Delta( y) = phi holds for any x, y is an element of C, x not equal y, where Delta( x) = {j - i vertical bar i, j is an element of x, i not equal j}. A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if boolean OR (x is an element of C) Delta( x) = Z(n) \ {0} holds and any codeword x is an element of C has the form {0, i, 2i,..., ( k - 1) i}. The concept of a CAC is motivated from applications in multiple- access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equidifference CACs ofweight k = 3 and characterize the code length n's admitting the condition through a number theoretical approach.
tight equi-difference conflict-avoiding codes 2(r)-th power residue kronecker density