A well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ring R . A sufficient condition is given, as well, for such a bound to be sharp. It is also shown that an MPC is free when its input codes are all free, in which case a generator matrix is given. If R is finite, a sufficient condition is provided for the dual of an MPC to be an MPC, a generator matrix for such a dual is given, and characterizations of LCD, self-dual, and self-orthogonal MPCs are presented. Finally, the results of this paper are used along with previous results of the authors to construct novel MPCs arising from σ , δ -codes. Some properties of such constructions are also studied.

Let H λ 4 be the Hecke group x , y : x 2 = y 4 = 1 and, for a square-free positive integer n , consider the subset ℚ ∗ − n = a + − n / c | a , b = a 2 + n / c ∈ ℤ , c ∈ 2 ℤ of the quadratic imaginary number field ℚ − n . Following a line of research in the relevant literature, we study the properties of the action of H λ 4 on ℚ ∗ − n . In particular, we calculate the number of orbits arising from this action for every such n . Some illustrative examples are also given.

Let K be a number field defined by a monic irreducible polynomial F(X) ∈ ℤ [X], p a fixed rational prime, and ν_p the discrete valuation associated to p. Assume that F(X) factors modulo p into the product of powers of r distinct monic irreducible polynomials. We present in this paper a condition, weaker than the known ones, which guarantees the existence of exactly r valuations of K extending ν_p. We further specify the ramification indices and residue degrees of these extended valuations in such a way that generalizes the known estimates. Some useful remarks and computational examples are also given to highlight some improvements due to our result.

Let \( (K,\nu) \) be an arbitrary-rank valued field, let \( R_{\nu} \) be the valuation ring of \( (K,\nu) \), and let \( K(\alpha)/K \) be a separable finite field extension generated over \( K \) by a root of a monic irreducible polynomial \( f\in R_{\nu}[X] \). We give some necessary and sufficient conditions for \( R_{\nu}[\alpha] \) to be integrally closed. We further characterize the integral closedness of \( R_{\nu}[\alpha] \) which is based on information about the valuations on \( K(\alpha) \) extending \( \nu \). Our results enhance and generalize some existing results as well as provide applications and examples.