Research Article
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Year 2020, Volume 28, Issue 28, 206 - 219, 14.07.2020
https://doi.org/10.24330/ieja.768265

Abstract

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
https://doi.org/10.24330/ieja.768265

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
Subjects Mathematics
Journal Section Articles
Authors

Virgilio P. SISON This is me (Primary Author)
University of the Philippines Los Banos
Philippines

Publication Date July 14, 2020
Published in Issue 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 <https://dergipark.org.tr/en/pub/ieja/issue/55997/768265>
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