(\sigma,\delta)-codes

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November 2013, 7(4): 463-474. doi: 10.3934/amc.2013.7.463

($\sigma,\delta$)-codes

M'Hammed Boulagouaz ^1, and André Leroy ^2,

1.	University of King Khalid, Abha, Saudi Arabia
2.	Université d'Artois, Lille Nord de France, Rue Jean Souvraz, Lens, 62300, France

Received October 2012 Published October 2013

Abstract
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In this paper we introduce the notion of cyclic ($f(t),\sigma,\delta$)-codes for $f(t)\in A[t;\sigma,\delta]$. These codes generalize the $\theta$-codes as introduced by D. Boucher, F. Ulmer, W. Geiselmann [2]. We construct generic and control matrices for these codes. As a particular case the ($\sigma,\delta$)-$W$-code associated to a Wedderburn polynomial are defined and we show that their control matrices are given by generalized Vandermonde matrices. All the Wedderburn polynomials of $\mathbb F_q[t;\theta]$ are described and their control matrices are presented. A key role will be played by the pseudo-linear transformations.

Keywords: Cyclic codes, Ore extensions, pseudo-linear transformations..

Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 16S3.

Citation: M'Hammed Boulagouaz, André Leroy. ($\sigma,\delta$)-codes. Advances in Mathematics of Communications, 2013, 7 (4) : 463-474. doi: 10.3934/amc.2013.7.463

References:

[1]	S. A. Amitsur, Derivations in simple rings, Proc. London Math. Soc., 3 (1957), 87-112.
[2]	D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engin. Commun. Comp., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z.
[3]	D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292. doi: 10.3934/amc.2008.2.273.
[4]	D. Boucher and F. Ulmer, Linear codes using skew polynomials with automorphisms and derivations, Des. Codes Cryptogr., to appear. doi: 10.1007/s10623-012-9704-4.
[5]	J. Delenclos and A. Leroy, Noncommutative symmetric functions and W-polynomials, J. Algebra Appl., 6 (2007), 815-837. doi: 10.1142/S021949880700251X.
[6]	M. Giesbrecht, Factoring in skew polynomial rings over finite fields, J. Symb. Comp., 26 (1998), 463-468. doi: 10.1006/jsco.1998.0224.
[7]	N. Jacobson, On pseudo linear transformations, Ann. Math., 38 (1937), 484-507. doi: 10.2307/1968565.
[8]	S. K. Jain and S. R. Nagpaul, Topics in Applied Abstract Algebra, AMS, 2005.
[9]	T. Y. Lam and A. Leroy, Wedderburn polynomials over division rings, I, J. Pure Appl. Algebra, 186 (2004), 43-76. doi: 10.1016/S0022-4049(03)00125-7.
[10]	T. Y. Lam, A. Leroy and A. Ozturk, Wedderburn polynomial over division rings, II, Contemp. Math., 456 (2008), 73-98. doi: 10.1090/conm/456/08885.
[11]	A. Leroy, Pseudo-linear transformation and evaluation in Ore extension, Bull. Belg. Math. Soc., 2 (1995), 321-345.
[12]	A. Leroy, Noncommutative polynomial maps, J. Algebra Appl., 11 (2012). doi: 10.1142/S0219498812500764.
[13]	S. R. López-Permouth and S. Szabo, Convolutional codes with additional algebraic structures, J. Pure Appl. Algebra, (2012). doi: 10.1016/j.jpaa.2012.09.017.
[14]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1978.
[15]	P. Solé, Codes over rings, in Proceeding of the CIMPA Summer School, Ankara, Turkey, 2008. doi: 10.1109/TIT.2013.2277721.
[16]	P. Solé and O. Yemen, Binary quasi-cyclic codes of index 2 and skew polynomial rings, Finite Fields Appl., 18 (2012), 685-699. doi: 10.1016/j.ffa.2012.02.002.
[17]	J. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc., 136 (2008), 699-706. doi: 10.1090/S0002-9939-07-09164-2.

show all references

References:

[1]	S. A. Amitsur, Derivations in simple rings, Proc. London Math. Soc., 3 (1957), 87-112.
[2]	D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engin. Commun. Comp., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z.
[3]	D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292. doi: 10.3934/amc.2008.2.273.
[4]	D. Boucher and F. Ulmer, Linear codes using skew polynomials with automorphisms and derivations, Des. Codes Cryptogr., to appear. doi: 10.1007/s10623-012-9704-4.
[5]	J. Delenclos and A. Leroy, Noncommutative symmetric functions and W-polynomials, J. Algebra Appl., 6 (2007), 815-837. doi: 10.1142/S021949880700251X.
[6]	M. Giesbrecht, Factoring in skew polynomial rings over finite fields, J. Symb. Comp., 26 (1998), 463-468. doi: 10.1006/jsco.1998.0224.
[7]	N. Jacobson, On pseudo linear transformations, Ann. Math., 38 (1937), 484-507. doi: 10.2307/1968565.
[8]	S. K. Jain and S. R. Nagpaul, Topics in Applied Abstract Algebra, AMS, 2005.
[9]	T. Y. Lam and A. Leroy, Wedderburn polynomials over division rings, I, J. Pure Appl. Algebra, 186 (2004), 43-76. doi: 10.1016/S0022-4049(03)00125-7.
[10]	T. Y. Lam, A. Leroy and A. Ozturk, Wedderburn polynomial over division rings, II, Contemp. Math., 456 (2008), 73-98. doi: 10.1090/conm/456/08885.
[11]	A. Leroy, Pseudo-linear transformation and evaluation in Ore extension, Bull. Belg. Math. Soc., 2 (1995), 321-345.
[12]	A. Leroy, Noncommutative polynomial maps, J. Algebra Appl., 11 (2012). doi: 10.1142/S0219498812500764.
[13]	S. R. López-Permouth and S. Szabo, Convolutional codes with additional algebraic structures, J. Pure Appl. Algebra, (2012). doi: 10.1016/j.jpaa.2012.09.017.
[14]	F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1978.
[15]	P. Solé, Codes over rings, in Proceeding of the CIMPA Summer School, Ankara, Turkey, 2008. doi: 10.1109/TIT.2013.2277721.
[16]	P. Solé and O. Yemen, Binary quasi-cyclic codes of index 2 and skew polynomial rings, Finite Fields Appl., 18 (2012), 685-699. doi: 10.1016/j.ffa.2012.02.002.
[17]	J. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc., 136 (2008), 699-706. doi: 10.1090/S0002-9939-07-09164-2.

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