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Year 2020, , 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, , 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.
There are 10 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Virgilio P. Sıson This is me

Publication Date July 14, 2020
Published in Issue Year 2020

Cite

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. https://doi.org/10.24330/ieja.768265
AMA Sıson VP. BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$. IEJA. July 2020;28(28):206-219. doi:10.24330/ieja.768265
Chicago 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, no. 28 (July 2020): 206-19. https://doi.org/10.24330/ieja.768265.
EndNote Sıson VP (July 1, 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.
IEEE V. P. Sıson, “BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$”, IEJA, vol. 28, no. 28, pp. 206–219, 2020, doi: 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.
JAMA Sıson VP. BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$. IEJA. 2020;28:206–219.
MLA 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, vol. 28, no. 28, 2020, pp. 206-19, doi:10.24330/ieja.768265.
Vancouver Sıson VP. BASES AND AUTOMORPHISM MATRIX OF THE GALOIS RING $GR(p^r,m)$ OVER $\mathbb{Z}_{p^r}$. IEJA. 2020;28(28):206-19.