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

($\sigma,\delta$)-codes

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.
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 derivationsDes. 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 derivationsDes. 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.

[1]

İsmail Aslan, Türkan Yeliz Gökçer. Approximation by pseudo-linear discrete operators. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021037

[2]

Sara D. Cardell, Joan-Josep Climent, Daniel Panario, Brett Stevens. A construction of $ \mathbb{F}_2 $-linear cyclic, MDS codes. Advances in Mathematics of Communications, 2020, 14 (3) : 437-453. doi: 10.3934/amc.2020047

[3]

Yun Gao, Shilin Yang, Fang-Wei Fu. Some optimal cyclic $ \mathbb{F}_q $-linear $ \mathbb{F}_{q^t} $-codes. Advances in Mathematics of Communications, 2021, 15 (3) : 387-396. doi: 10.3934/amc.2020072

[4]

Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028

[5]

Nabil Bennenni, Kenza Guenda, Sihem Mesnager. DNA cyclic codes over rings. Advances in Mathematics of Communications, 2017, 11 (1) : 83-98. doi: 10.3934/amc.2017004

[6]

Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177

[7]

Pavel Galashin, Vladimir Zolotov. Extensions of isometric embeddings of pseudo-Euclidean metric polyhedra. Electronic Research Announcements, 2016, 23: 1-7. doi: 10.3934/era.2016.23.001

[8]

Mahesh Nerurkar. Forced linear oscillators and the dynamics of Euclidean group extensions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1201-1234. doi: 10.3934/dcdss.2016049

[9]

Heide Gluesing-Luerssen, Fai-Lung Tsang. A matrix ring description for cyclic convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 55-81. doi: 10.3934/amc.2008.2.55

[10]

Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025

[11]

Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017

[12]

Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018

[13]

Heide Gluesing-Luerssen, Uwe Helmke, José Ignacio Iglesias Curto. Algebraic decoding for doubly cyclic convolutional codes. Advances in Mathematics of Communications, 2010, 4 (1) : 83-99. doi: 10.3934/amc.2010.4.83

[14]

San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038

[15]

Yan Liu, Xiwang Cao, Wei Lu. Two classes of new optimal ternary cyclic codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021033

[16]

Heide Gluesing-Luerssen, Hunter Lehmann. Automorphism groups and isometries for cyclic orbit codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021040

[17]

Enhui Lim, Frédérique Oggier. On the generalised rank weights of quasi-cyclic codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022010

[18]

Gustavo Terra Bastos, Reginaldo Palazzo Júnior, Marinês Guerreiro. Abelian non-cyclic orbit codes and multishot subspace codes. Advances in Mathematics of Communications, 2020, 14 (4) : 631-650. doi: 10.3934/amc.2020035

[19]

Martianus Frederic Ezerman, San Ling, Patrick Solé, Olfa Yemen. From skew-cyclic codes to asymmetric quantum codes. Advances in Mathematics of Communications, 2011, 5 (1) : 41-57. doi: 10.3934/amc.2011.5.41

[20]

Yunwen Liu, Longjiang Qu, Chao Li. New constructions of systematic authentication codes from three classes of cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 1-16. doi: 10.3934/amc.2018001

2020 Impact Factor: 0.935

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]