We discuss self-dual codes over the ring [Formula: see text]. We characterize the structure of self-dual, Type I codes and Type II codes over [Formula: see text] with given generator matrix in terms of the structure of their Torsion and Residue codes. We construct self-dual, Type I and Type II codes over [Formula: see text] for different lengths.
Binary self-dual codes and additive self-dual codes over GF(4) contain common points. Both have Type I codes and Type II codes, as well as shadow codes. In this paper, we provide a comprehensive description of extremal and near-extremal Type I codes over GF(2) and GF(4) with minimal shadow. In particular, we prove that there is no near-extremal Type I [24m,12m,2m+2] binary self-dual code with minimal shadow if m≥323, and we prove that there is no near-extremal Type I (6m+1,26m+1,2m+1) additive self-dual code over GF(4) with minimal shadow if m≥22.
In this paper we study linear codes over the rings Ri = ℤ4 + wℤ4, where w2 = i, i = 1, 2w. We characterize self-dual codes over Ri and propose a construction method for self-dual codes over these rings. We also briefly discuss circulant self-dual codes and Type II codes. Some examples are given.
For a positive integer m, let R be either the ring ℤ2m of integers modulo 2m or the quaternionic ring Σ2m = ℤ2m + αℤ2m + βℤ2m + γℤ2m with α = 1 + î, β = 1 + ĵ and [Formula: see text], where [Formula: see text] are elements of the ring ℍ of real quaternions satisfying [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we obtain Jacobi forms (or Siegel modular forms) of genus r from byte weight enumerators (or symmetrized byte weight enumerators) in genus r of Type I and Type II codes over R. Furthermore, we derive a functional equation for partial Epstein zeta functions, which are summands of classical Epstein zeta functions associated with quadratic forms.
Configurations of Extremal Type II Codes via Harmonic Weight Enumerators