On Perfect Domination of q-Ary Cubes

The construction of perfect dominating sets in all binary 2-cubes (r > 3) with the same number of edges along each coordinate direction is performed, thus generalizing the known construction of this for r = 3. However, an adaptation of that construction aiming to perfect dominating sets in the 14-ternary cube with edges along all coordinate directions still fails. 1 Perfect Dominating Sets in Binary Cubes Given a graph G, a perfect dominating set, or PDS, of G is a vertex subset S such that every vertex of G not in S is a neighbor of exactly one vertex of S. In this section we generalize a construction in [1] of a PDS in the binary 8-cube inducing linear components in all coordinate directions. Let g1(x) and g2(x) be two primitive binary polynomials of degree r and let C1 and C2 be the cyclic Hamming codes of length 2 −1 generated by these polynomials. The cyclic code C = C1 ∩ C2 has g1(x)g2(x) as its generator polynomial. Form the extended code C 1 of length 2 r (by adding an overall parity-check digit). Form the extended code C 2 by adding the complement