BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$

Abstract

Let $GR(p^r,m)$ denote the Galois ring of characteristic $p^r$ and cardinality $p^{rm}$ seen as a free module of rank $m$ over the integer ring $\mathbb{Z}_{p^r}$. A general formula for the sum of the homogeneous weights of the $p^r$-ary images of elements of $GR(p^r,m)$ under any basis is derived in terms of the parameters of $GR(p^r,m)$. By using a Vandermonde matrix over $GR(p^r,m)$ with respect to the generalized Frobenius automorphism, a constructive proof that every basis of $GR(p^r,m)$ has a unique dual basis is given. It is shown that a basis is self-dual if and only if its automorphism matrix is orthogonal, and that a basis is normal if and only if its automorphism matrix is symmetric.

Keywords

• Galois ring
• Vandermonde matrix
• dual basis
• normal basis

References

1. I. Constantinescu, W. Heise and T. Honold, Monomial extensions of isometries between codes over $\mathbb{Z}_M$, Proceedings of the 5th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT '96), Unicorn Shumen, (1996), 98-104.
2. M. Greferath and S. E. Schmidt, Gray isometries for finite chain rings and a nonlinear ternary $(36,3^{12},15)$ code, IEEE Trans. Inform. Theory, 45(7) (1999), 2522-2524.
3. M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams' Equivalence Theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28.
4. T. Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel), 76 (2001), 406-415.
5. B. R. MacDonald, Finite Rings with Identity, Pure and Applied Mathematics, 28, Marcel Dekker, Inc., New York, 1974.
6. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, I, North-Holland Mathematical Library, 16, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
7. A. A. Nechaev and T. Honold, Fully weighted modules and representations of codes, Problems Inform. Transmission, 35(3) (1999), 205-223.
8. P. Rabizzoni, Relation between the minimum weight of a linear code over $GF(q^m)$ and its q-ary image over $GF(q)$, Coding theory and applications (Toulon, 1988), Lecture Notes in Comput. Sci., Springer, New York, 388 (1989), 209-212.

9. P. Sole and V. Sison, Bounds on the minimum homogeneous distance of the $p^r$-ary image of linear block codes over the Galois ring $GR(p^r,m)$, IEEE Trans. Inform. Theory, 53(6) (2007), 2270-2273.
10. Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.