American Institute of Mathematical Sciences

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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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Boucher and F. Ulmer, Linear codes using skew polynomials with automorphisms and derivations,, Des. Codes Cryptogr., ().  doi: 10.1007/s10623-012-9704-4.  Google Scholar

[5]

J. Delenclos and A. Leroy, Noncommutative symmetric functions and W-polynomials, J. Algebra Appl., 6 (2007), 815-837. doi: 10.1142/S021949880700251X.  Google Scholar

[6]

M. Giesbrecht, Factoring in skew polynomial rings over finite fields, J. Symb. Comp., 26 (1998), 463-468. doi: 10.1006/jsco.1998.0224.  Google Scholar

[7]

N. Jacobson, On pseudo linear transformations, Ann. Math., 38 (1937), 484-507. doi: 10.2307/1968565.  Google Scholar

[8]

S. K. Jain and S. R. Nagpaul, Topics in Applied Abstract Algebra, AMS, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Leroy, Pseudo-linear transformation and evaluation in Ore extension, Bull. Belg. Math. Soc., 2 (1995), 321-345.  Google Scholar

[12]

A. Leroy, Noncommutative polynomial maps, J. Algebra Appl., 11 (2012). doi: 10.1142/S0219498812500764.  Google Scholar

[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.  Google Scholar

[14]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1978. Google Scholar

[15]

P. Solé, Codes over rings, in Proceeding of the CIMPA Summer School, Ankara, Turkey, 2008. doi: 10.1109/TIT.2013.2277721.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. A. Amitsur, Derivations in simple rings, Proc. London Math. Soc., 3 (1957), 87-112.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Boucher and F. Ulmer, Linear codes using skew polynomials with automorphisms and derivations,, Des. Codes Cryptogr., ().  doi: 10.1007/s10623-012-9704-4.  Google Scholar

[5]

J. Delenclos and A. Leroy, Noncommutative symmetric functions and W-polynomials, J. Algebra Appl., 6 (2007), 815-837. doi: 10.1142/S021949880700251X.  Google Scholar

[6]

M. Giesbrecht, Factoring in skew polynomial rings over finite fields, J. Symb. Comp., 26 (1998), 463-468. doi: 10.1006/jsco.1998.0224.  Google Scholar

[7]

N. Jacobson, On pseudo linear transformations, Ann. Math., 38 (1937), 484-507. doi: 10.2307/1968565.  Google Scholar

[8]

S. K. Jain and S. R. Nagpaul, Topics in Applied Abstract Algebra, AMS, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Leroy, Pseudo-linear transformation and evaluation in Ore extension, Bull. Belg. Math. Soc., 2 (1995), 321-345.  Google Scholar

[12]

A. Leroy, Noncommutative polynomial maps, J. Algebra Appl., 11 (2012). doi: 10.1142/S0219498812500764.  Google Scholar

[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.  Google Scholar

[14]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1978. Google Scholar

[15]

P. Solé, Codes over rings, in Proceeding of the CIMPA Summer School, Ankara, Turkey, 2008. doi: 10.1109/TIT.2013.2277721.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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