Abstract — A length � group code over a group q is a subgroup of q � under component-wise group operation. Group codes over dihedral groups hw, with Pw elements, that are two-level constructible using a binary code and a code over w residue class integer ring modulo w, as component codes are studied for arbitrary w. A set of necessary and sufficient conditions on the component codes for the two-level construction to result in a group code over hw are obtained. The conditions differ for w odd and even. Using two-level group codes over hw as label codes, performance of block-coded modulation scheme is discussed under all possible matched labelings of Pw-APSK and Pw-SPSK (asymmetric and symmetric PSK) signal sets in terms of the minimum squared Euclidean distance. Matched labelings that lead to Automorphic Euclidean Distance Equivalent codes are identified. It is shown that depending upon the ratio of Hamming distances of the component codes some labelings perform better than other. The best labeling is identified under a set of restrictive conditions. Finally, conditions on the component codes for phase rotational invariance properties of the signal space codes are discussed. Index Terms—Coded modulation, dihedral groups, group codes, multilevel codes. I.

Codes over Fq m that are closed under addition, and multiplication with elements from Fq are called Fq -linear codes over Fq m . For m 6= 1, this class of codes is a subclass of nonlinear codes. Among Fq -linear codes, we consider only cyclic codes and call them Fq -linear cyclic codes (FqLC codes) over Fq m . The class of FqLC codes includes as special cases (i) group cyclic codes over elementary Abelian groups (q = p, a prime), (ii) subspace subcodes of Reed-Solomon codes (n = q 1) studied by Hattori, McEliece and Solomon (iii) linear cyclic codes over Fq (m=1) and (iv) twisted BCH codes. Moreover, with respect to any particular Fq -basis of Fq m , any FqLC code over Fq m can be viewed as an m-quasi-cyclic code of length mn over Fq . In this correspondence, we obtain transform domain characterization of FqLC codes, using Discrete Fourier Transform (DFT) over an extension eld of Fq m . The characterization is in terms of any decomposition of the code into certain subcodes and linearized polynomials over Fq m . We show how one can use this transform domain characterization to obtain a minimum distance bound for the corresponding quasi-cyclic code. We also prove nonexistence of self dual FqLC codes and self dual quasi-cyclic codes of certain parameters using the transform domain characterization.

Codes over Fq that form vector spaces over Fq are called Fq-linear codes over Fq. Among these we consider only cyclic codes and call them Fq-linear cyclic codes (FqLC codes) over Fq. This class of codes includes as special cases (i) group cyclic codes over elementary abelian groups (q = p, a prime), (ii) subspace subcodes of Reed-Solomon codes and (iii) linear cyclic codes over Fq (re=l). Transform domain characterization of FqLC codes is obtained using Discrete Fourier Transform (DFT) over an extension field of Fq. We show how one can use this transform domain structures to estimate a minimum distance bound for the corresponding quasicyclic code by BOll-like argument.

In this paper we present a matrix characterization of AMDS codes and NMDS codes over Zm and Abelian groups.

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