Let G be a group. A k-coloring of G is a surjection λ:G→{1,2,…,k}. Equivalently, a k-coloring λ of G is a partition P={P1,P2,…,Pk} of G into k subsets. If gP=P for all g in G, we say that λ is perfect. If hP=P only for all h∈H≤G such that [G:H]=2, then λ is semiperfect. If there is an element g∈G such that λ(x)=λ(gx−1g) for all x∈G, then λ is said to be symmetric. In this research, we relate the notion of symmetric colorings with perfect and semiperfect colorings. Specifically, we identify which perfect and semiperfect colorings are symmetric in relation to the subgroups of G that contain the squares of elements in G, in H, and in G∖H. We also show examples of colored planar patterns that represent symmetric perfect and symmetric semiperfect colorings of some groups.

Cyclic and self-dual codes are important classes of codes in coding theory. Jia, Ling and Xing [5] as well as Kai and Zhu [7] proved that Euclidean self-dual cyclic codes of length n over Fq exist if and only if n is even and q = 2 r , where r is any positive integer. For n and q even, there always exists an [n, n 2 ] self-dual cyclic code with generator polynomial x n 2 + 1 called the trivial self-dual cyclic code. In this paper we prove the existence of nontrivial self-dual cyclic codes of length n = 2 ν•n, where n is odd, over F2r in terms of the existence of a nontrivial splitting (Z, X0, X1) of Z n by µ−1, where Z, X0, X1 are unions of 2 r -cyclotomic cosets mod n. We also express the formula for the number of cyclic self-dual codes over F2r for each n and r in terms of the number of 2 r -cyclotomic cosets in X0 (or in X1).We also look at Hermitian self-dual cyclic codes and show properties which are analogous to those of Euclidean self-dual cyclic codes. That is, the existence of nontrivial Hermitian self-dual codes over F 2 2ℓ based on the existence of a nontrivial splitting (Z, X0, X1) of Zn by µ −2 ℓ , where Z, X0, X1 are unions of 2 2ℓ -cyclotomic cosets mod n. We also determine the lengths at which nontrivial Hermitian self-dual cyclic codes exist and the formula for the number of Hermitian self-dual cyclic codes for each n.

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