Let p be an odd prime and Λ=Z[D2p] the integral group ring of dihedral D2p order 2p. The syzygies Ωr(Z) are stable classes intermediate modules in a free Λ-resolution trivial module. We will discuss explicitly interaction under − ⊗Z−.

‎in this paper‎, ‎we obtain the dimensions of symmetry classes of polynomials associated with‎ ‎the irreducible characters of the dihedral group as a subgroup of‎ ‎the full symmetric group‎. ‎then we discuss the existence of o-basis‎ ‎of these classes‎.

Dihedral codes, particular cases of quasi-cyclic have a nice algebraic structure which allows to store them efficiently. In this paper, we investigate it and prove some lower bounds on their dimension minimum distance, in analogy with the theory BCH codes. This us construct dihedral codes prescribed distance. binary case, present examples optimal obtained by construction.