In this correspondence, bounds are derived on the minimum homogeneous distance of the image of a linear block code over the Galois ring GR(pr,m), with respect to any basis of GR(pr,m). These bounds depend on the parameters of GR(pr ,m), the minimum Hamming distance of the block code, and the average value of the homogeneous weight applied on the base ring Zopf p r. Examples are given of Galois ring codes that meet these bounds

Bounds on the Minimum Homogeneous Distance of the ${p^{r}}$ -ary Image of Linear Block Codes Over the Galois Ring ${\hbox{GR}}(p^{r},m)$

In this paper, bounds are derived on the minimum homogeneous distance of the image of a linear block code over the Galois ring GR(pr ,m), with respect to any basis of GR(pr, m) These bounds depend on the parameters of GR(pr , m), the minimum Hamming distance of the block code, and the average value of the homogeneous weight applied on the base ring Zpr Examples are given of Galois ring codes that meet these bounds

Bounds on the Minimum Homogeneous Distance of the p r -ary Image of Linear Block Codes over the Galois Ring GR(p r , m)

This paper introduces for the first time a characterization of the q-ary images of qn-ary σ-codes, i.e. linear codes of length m over F q n that are invariant under the action of a permutation σ, in the special case when F q n = F q (β) = F q (βl) for some l > 0, β ∊ F q n. This is achieved by introducing an appropriate module structure on F q nm.

An algebraic characterization of q-ary images of q n -ary codes invariant under a permutation

The Galois ring $GR(p^r,m)$ of characteristic $p^r$ and cardinality $p^{rm}$, where $p$ is a prime and $r,m \ge 1$ are integers, is a Galois extension of the residue class ring $Z_{p^r}$ by a root $\omega$ of a monic basic irreducible polynomial of degree $m$ over $Z_{p^r}$. Every element of $GR(p^r,m)$ can be expressed uniquely as a polynomial in $\omega$ with coefficients in $Z_{p^r}$ and degree less than or equal to $m-1$, thus $GR(p^r,m)$ is a free module of rank $m$ over $Z_{p^r}$ with basis $\{1,\omega, \omega^2,..., \omega^{m-1} \}$. The ring $Z_{p^r}$ satisfies the invariant dimension property, hence any other basis of $GR(p^r,m)$, if it exists, will have cardinality $m$. 
This paper was motivated by the code-theoretic problem of finding the homogeneous bound on the $p^r$-image of a linear block code over $GR(p^r,m)$ with respect to any basis. It would be interesting to consider the dual and normal bases of $GR(p^r,m)$. 
By using a Vandermonde matrix over $GR(p^r,m)$ in terms of the generalized Frobenius automorphism, a constructive proof that every basis of $GR(p^r,m)$ has a unique dual basis is given. The notion of normal bases was also generalized from the classic case for Galois fields.

Bases of the Galois Ring $GR(p^r,m)$ over the Integer Ring $Z_{p^r}$

Galois ring codes and their images under various bases

In this paper we give an upper bound for the minimum distance of the q-ary image of a linear code over GF(qs) This allows us to give some indications about the minimum distance of the q-ary image of MDS codes over GF(qs) when q=2 and q=3

Patrice Rabizzoni

Papers

Relation between the minimum weight of a linear code over GF(qm) and its q-art image over GF(q)