Some Applications of the Fourier Transform in Algebraic Coding Theory
This expository article describes two uses of the Fourier transform of interest in algebraic coding theory: the MacWilliams identities on weight enumerators of linear codes and the decomposition of a semi-simple group algebra of a finite group into a direct sum of matrix rings.
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| Vydané v: | Algebra for Secure and Reliable Communication Modeling Ročník 642; s. 1 - 40 |
|---|---|
| Hlavný autor: | |
| Médium: | Kapitola |
| Jazyk: | English |
| Vydavateľské údaje: |
Providence, Rhode Island
American Mathematical Society
23.06.2015
|
| Edícia: | Contemporary Mathematics |
| ISBN: | 1470410184, 9781470410186 |
| ISSN: | 0271-4132, 1098-3627 |
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| Abstract | This expository article describes two uses of the Fourier transform of interest in algebraic coding theory: the MacWilliams
identities on weight enumerators of linear codes and the decomposition of a semi-simple group algebra of a finite group into a
direct sum of matrix rings. |
|---|---|
| AbstractList | This expository article describes two uses of the Fourier transform of interest in algebraic coding theory: the MacWilliams
identities on weight enumerators of linear codes and the decomposition of a semi-simple group algebra of a finite group into a
direct sum of matrix rings. |
| Author | Wood, Jay A. |
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| Editor | Lahyane, Mustapha Martínez-Moro, Edgar |
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| References | Gabriele Nebe, Eric M. Rains, and Neil J. A. Sloane, Self-dual codes and invariant theory, Algorithms and Computation in Mathematics, vol. 17, Springer-Verlag, Berlin, 2006. MR 2209183 (2007d:94066) V. A. Zinov′ev and T. Èrikson, On Fourier-invariant partitions of finite abelian groups and on the MacWilliams identity for group codes, Problemy Peredachi Informatsii 32 (1996), no. 1, 137–143 (Russian, with Russian summary); English transl., Problems Inform. Transmission 32 (1996), no. 1, 117–122. MR 1384939 (97m:20062) A. Roger Hammons Jr., P. Vijay Kumar, A. R. Calderbank, N. J. A. Sloane, and Patrick Solé, The Z4\textbf {Z}_4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40 (1994), no. 2, 301–319. MR 1294046 (95k:94030), DOI 10.1109/18.312154 Thomas Honold and Ivan Landjev, MacWilliams identities for linear codes over finite Frobenius rings, Finite fields and applications (Augsburg, 1999) Springer, Berlin, 2001, pp. 276–292. MR 1849094 (2002i:94066) T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294 (99i:16001), DOI 10.1007/978-1-4612-0525-8 Heide Gluesing-Luerssen, Partitions of Frobenius rings induced by the homogeneous weight, Adv. Math. Commun. 8 (2014), no. 2, 191–207. MR 3209298, DOI 10.3934/amc.2014.8.191 Jay A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555–575. MR 1738408 (2001d:94033) V. A. Zinov′ev and T. Èrikson, Fourier-invariant pairs of partitions of finite abelian groups, and association schemes, Problemy Peredachi Informatsii 45 (2009), no. 3, 33–44 (Russian, with Russian summary); English transl., Probl. Inf. Transm. 45 (2009), no. 3, 221–231. MR 2590742 (2010k:94008), DOI 10.1134/S003294600903003X Audrey Terras, Fourier analysis on finite groups and applications, London Mathematical Society Student Texts, vol. 43, Cambridge University Press, Cambridge, 1999. MR 1695775 (2000d:11003), DOI 10.1017/CBO9780511626265 Thomas Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel) 76 (2001), no. 6, 406–415. MR 1831096 (2002b:16033), DOI 10.1007/PL00000451 Heinrich Maschke, Ueber den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen, Math. Ann. 50 (1898), no. 4, 492–498 (German). MR 1511011, DOI 10.1007/BF01444297 A. Fröhlich and M. J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, Cambridge, 1993. MR 1215934 (94d:11078) I. Konstantinesku and V. Khaĭze, A metric for codes over residue class rings of integers, Problemy Peredachi Informatsii 33 (1997), no. 3, 22–28 (Russian, with Russian summary); English transl., Problems Inform. Transmission 33 (1997), no. 3, 208–213 (1998). MR 1476368 (99a:94058) Y. Hirano, On admissible rings, Indag. Math. (N.S.) 8 (1997), no. 1, 55–59. MR 1617802 (99b:16034), DOI 10.1016/S0019-3577(97)83350-2 Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380 (56 \#8675) Jay A. Wood, Anti-isomorphisms, character modules and self-dual codes over non-commutative rings, Int. J. Inf. Coding Theory 1 (2010), no. 4, 429–444. MR 2772908 (2011m:94134), DOI 10.1504/IJICOT.2010.032867 Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR 0144979 (26 \#2519) Eimear Byrne, Marcus Greferath, and Michael E. O’Sullivan, The linear programming bound for codes over finite Frobenius rings, Des. Codes Cryptogr. 42 (2007), no. 3, 289–301. MR 2298938 (2008c:94053), DOI 10.1007/s10623-006-9035-4 D. J. Benson, Representations and cohomology. I. Basic representation theory of finite groups and associative algebras, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998. MR 1644252 (99f:20001a) |
| References_xml | – reference: A. Fröhlich and M. J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, Cambridge, 1993. MR 1215934 (94d:11078) – reference: Heinrich Maschke, Ueber den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen, Math. Ann. 50 (1898), no. 4, 492–498 (German). MR 1511011, DOI 10.1007/BF01444297 – reference: Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott; Graduate Texts in Mathematics, Vol. 42. MR 0450380 (56 \#8675) – reference: Jay A. Wood, Anti-isomorphisms, character modules and self-dual codes over non-commutative rings, Int. J. Inf. Coding Theory 1 (2010), no. 4, 429–444. MR 2772908 (2011m:94134), DOI 10.1504/IJICOT.2010.032867 – reference: Audrey Terras, Fourier analysis on finite groups and applications, London Mathematical Society Student Texts, vol. 43, Cambridge University Press, Cambridge, 1999. MR 1695775 (2000d:11003), DOI 10.1017/CBO9780511626265 – reference: Y. Hirano, On admissible rings, Indag. Math. (N.S.) 8 (1997), no. 1, 55–59. MR 1617802 (99b:16034), DOI 10.1016/S0019-3577(97)83350-2 – reference: V. A. Zinov′ev and T. Èrikson, Fourier-invariant pairs of partitions of finite abelian groups, and association schemes, Problemy Peredachi Informatsii 45 (2009), no. 3, 33–44 (Russian, with Russian summary); English transl., Probl. Inf. Transm. 45 (2009), no. 3, 221–231. MR 2590742 (2010k:94008), DOI 10.1134/S003294600903003X – reference: Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. MR 0144979 (26 \#2519) – reference: V. A. Zinov′ev and T. Èrikson, On Fourier-invariant partitions of finite abelian groups and on the MacWilliams identity for group codes, Problemy Peredachi Informatsii 32 (1996), no. 1, 137–143 (Russian, with Russian summary); English transl., Problems Inform. Transmission 32 (1996), no. 1, 117–122. MR 1384939 (97m:20062) – reference: T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics, vol. 189, Springer-Verlag, New York, 1999. MR 1653294 (99i:16001), DOI 10.1007/978-1-4612-0525-8 – reference: Gabriele Nebe, Eric M. Rains, and Neil J. A. Sloane, Self-dual codes and invariant theory, Algorithms and Computation in Mathematics, vol. 17, Springer-Verlag, Berlin, 2006. MR 2209183 (2007d:94066) – reference: A. Roger Hammons Jr., P. Vijay Kumar, A. R. Calderbank, N. J. A. Sloane, and Patrick Solé, The Z4\textbf {Z}_4-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40 (1994), no. 2, 301–319. MR 1294046 (95k:94030), DOI 10.1109/18.312154 – reference: I. Konstantinesku and V. Khaĭze, A metric for codes over residue class rings of integers, Problemy Peredachi Informatsii 33 (1997), no. 3, 22–28 (Russian, with Russian summary); English transl., Problems Inform. Transmission 33 (1997), no. 3, 208–213 (1998). MR 1476368 (99a:94058) – reference: Heide Gluesing-Luerssen, Partitions of Frobenius rings induced by the homogeneous weight, Adv. Math. Commun. 8 (2014), no. 2, 191–207. MR 3209298, DOI 10.3934/amc.2014.8.191 – reference: Thomas Honold, Characterization of finite Frobenius rings, Arch. Math. (Basel) 76 (2001), no. 6, 406–415. MR 1831096 (2002b:16033), DOI 10.1007/PL00000451 – reference: Thomas Honold and Ivan Landjev, MacWilliams identities for linear codes over finite Frobenius rings, Finite fields and applications (Augsburg, 1999) Springer, Berlin, 2001, pp. 276–292. MR 1849094 (2002i:94066) – reference: D. J. Benson, Representations and cohomology. I. Basic representation theory of finite groups and associative algebras, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1998. MR 1644252 (99f:20001a) – reference: Eimear Byrne, Marcus Greferath, and Michael E. O’Sullivan, The linear programming bound for codes over finite Frobenius rings, Des. Codes Cryptogr. 42 (2007), no. 3, 289–301. MR 2298938 (2008c:94053), DOI 10.1007/s10623-006-9035-4 – reference: Jay A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), no. 3, 555–575. MR 1738408 (2001d:94033) |
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identities on weight enumerators of... |
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| Title | Some Applications of the Fourier Transform in Algebraic Coding Theory |
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