Research Article
Year 2020, Volume 28, Issue 28, 206 - 219, 14.07.2020

### References

• 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.
• 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.
• M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams' Equivalence Theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28.
• T. Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel), 76 (2001), 406-415.
• B. R. MacDonald, Finite Rings with Identity, Pure and Applied Mathematics, 28, Marcel Dekker, Inc., New York, 1974.
• 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.
• A. A. Nechaev and T. Honold, Fully weighted modules and representations of codes, Problems Inform. Transmission, 35(3) (1999), 205-223.
• 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.
• 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.
• Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.

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

Year 2020, Volume 28, Issue 28, 206 - 219, 14.07.2020

### 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.

### References

• 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.
• 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.
• M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams' Equivalence Theorem, J. Combin. Theory Ser. A, 92 (2000), 17-28.
• T. Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel), 76 (2001), 406-415.
• B. R. MacDonald, Finite Rings with Identity, Pure and Applied Mathematics, 28, Marcel Dekker, Inc., New York, 1974.
• 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.
• A. A. Nechaev and T. Honold, Fully weighted modules and representations of codes, Problems Inform. Transmission, 35(3) (1999), 205-223.
• 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.
• 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.
• Z.-X. Wan, Lectures on Finite Fields and Galois Rings, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.

### Details

Primary Language English Mathematics Articles Virgilio P. SISON This is me (Primary Author) University of the Philippines Los Banos Philippines July 14, 2020 Year 2020, Volume 28, Issue 28

### Cite

 Bibtex @research article { ieja768265, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2020}, volume = {28}, number = {28}, pages = {206 - 219}, doi = {10.24330/ieja.768265}, title = {BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING \$GR(p\^r,m)\$ OVER \$\\mathbb\{Z\}\_\{p\^r\}\$}, key = {cite}, author = {Sıson, Virgilio P.} } APA Sıson, V. P. (2020). BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$ . International Electronic Journal of Algebra , 28 (28) , 206-219 . DOI: 10.24330/ieja.768265 MLA Sıson, V. P. "BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$" . International Electronic Journal of Algebra 28 (2020 ): 206-219 Chicago Sıson, V. P. "BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$". International Electronic Journal of Algebra 28 (2020 ): 206-219 RIS TY - JOUR T1 - BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$ AU - Virgilio P.Sıson Y1 - 2020 PY - 2020 N1 - doi: 10.24330/ieja.768265 DO - 10.24330/ieja.768265 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 206 EP - 219 VL - 28 IS - 28 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.768265 UR - https://doi.org/10.24330/ieja.768265 Y2 - 2022 ER - EndNote %0 International Electronic Journal of Algebra BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$ %A Virgilio P. Sıson %T BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$ %D 2020 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 28 %N 28 %R doi: 10.24330/ieja.768265 %U 10.24330/ieja.768265 ISNAD Sıson, Virgilio P. . "BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$". International Electronic Journal of Algebra 28 / 28 (July 2020): 206-219 . https://doi.org/10.24330/ieja.768265 AMA Sıson V. P. BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$. IEJA. 2020; 28(28): 206-219. Vancouver Sıson V. P. BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$. International Electronic Journal of Algebra. 2020; 28(28): 206-219. IEEE V. P. Sıson , "BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$", International Electronic Journal of Algebra, vol. 28, no. 28, pp. 206-219, Jul. 2020, doi:10.24330/ieja.768265