Quaternions over Galois rings and their codes
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It is shown in this paper that, if R is a Frobenius ring, then the quaternion ring ℋ_a,b(R) is a Frobenius ring for all units a,b ∈ R. In particular, if q is an odd prime power then ℋ_a,b(𝔽_q) is the semisimple non-commutative matrix ring M_2(𝔽_q). Consequently, a homogeneous weight that depends on the field size q is obtained. On the other hand, the homogeneous weight of a finite Frobenius ring with a unique minimal ideal is derived in terms of the size of the ideal. This is illustrated by the quaternions over the Galois ring GR(2^r,m). Finally, one-sided linear block codes over the quaternions over Galois rings are constructed, and certain bounds on the homogeneous distance of the images of these codes are proved. These bounds are based on the Hamming distance of the quaternion code and the parameters of the Galois ring. Good examples of one-sided rate-2/6, 3-quasi-cyclic quaternion codes and their images are generated. One of these codes meets the Singleton bound and is therefore a maximum distance separable code.
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