November  2016, 10(4): 765-795. doi: 10.3934/amc.2016040

Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$

1. 

IRMAR (UMR 6625), Université de Rennes 1, Campus de Beaulieu, F-35042 Rennes

Received  December 2014 Revised  February 2016 Published  November 2016

The aim of this text is to construct and to enumerate self-dual $\theta$-cyclic and $\theta$-negacyclic codes over $\mathbb{F}_{p^2}$ where $p$ is a prime number and $\theta$ is the Frobenius automorphism.
Citation: Delphine Boucher. Construction and number of self-dual skew codes over $\mathbb{F}_{p^2}$. Advances in Mathematics of Communications, 2016, 10 (4) : 765-795. doi: 10.3934/amc.2016040
References:
[1]

G. K. Bakshi and M. Raka, Self-dual and self-orthogonal negacyclic codes of length $2p^n$ over a finite field,, Finite Fields Appl., 19 (2013), 39.  doi: 10.1016/j.ffa.2012.10.003.  Google Scholar

[2]

D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes,, Appl. Algebra Engin. Commun. Comp., 18 (2007), 379.  doi: 10.1007/s00200-007-0043-z.  Google Scholar

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D. Boucher and F. Ulmer, Self-dual skew codes and factorization of skew polynomials,, J. Symb. Comp., 60 (2014), 47.  doi: 10.1016/j.jsc.2013.10.003.  Google Scholar

[4]

X. Caruso and J. Leborgne, Some algorithms for skew polynomials over finite fields,, preprint, ().   Google Scholar

[5]

H. Q. Dinh, Repeated-root constacyclic codes of length $2 p^s$,, Finite Fields Appl., 18 (2012), 133.  doi: 10.1016/j.ffa.2011.07.003.  Google Scholar

[6]

J. von zur Gathen and J. Gerhard, Modern Computer Algebra,, Cambridge Univ. Press, (2013).  doi: 10.1017/CBO9781139856065.  Google Scholar

[7]

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

[8]

K. Guenda and T. A. Gulliver, Self-dual repeated root cyclic and negacyclic codes over finite fields,, in 2012 IEEE Int. Symp. Inform. Theory Proc., (2012), 2904.   Google Scholar

[9]

S. Han, J.-L. Kim, H. Lee and Y. Lee, Construction of quasi-cyclic self-dual codes,, Finite Fields Appl., 18 (2012), 613.  doi: 10.1016/j.ffa.2011.12.006.  Google Scholar

[10]

N. Jacobson, The Theory of Rings,, Amer. Math. Soc., (1943).   Google Scholar

[11]

S. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields,, IEEE Trans. Inform. Theory, 57 (2011), 2243.  doi: 10.1109/TIT.2010.2092415.  Google Scholar

[12]

X. Kai and S. Zhu, On cyclic self-dual codes,, Appl. Algebra Engin. Commun. Comp., 19 (2008), 509.  doi: 10.1007/s00200-008-0086-9.  Google Scholar

[13]

R. Lidl and H. Niederreiter, Finite Fields,, Cambridge Univ. Press, (1997).   Google Scholar

[14]

S. Ling, H. Niederreiter and P. Solé, On the algebraic structure of quasi-cyclic codes IV: Repeated roots Chain rings,, Des. Codes Crypt., 38 (2006), 337.  doi: 10.1007/s10623-005-1431-7.  Google Scholar

[15]

S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes I. Finite fields,, IEEE Trans. Inform. Theory, 47 (2001), 2751.  doi: 10.1109/18.959257.  Google Scholar

[16]

R. W. K. Odoni, On additive polynomials over a finite field,, Proc. Edinburgh Math. Soc., 42 (1999), 1.  doi: 10.1017/S0013091500019970.  Google Scholar

[17]

O. Ore, Theory of Non-Commutative Polynomials,, Ann. Math., 34 (1933), 480.  doi: 10.2307/1968173.  Google Scholar

[18]

A. Sahni and P. T. Sehgal, Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields,, Adv. Math. Commun., 9 (2015), 437.  doi: 10.3934/amc.2015.9.437.  Google Scholar

[19]

I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length,, Int. J. Inf. Coding Theory, 2 (2011), 10.  doi: 10.1504/IJICOT.2011.044674.  Google Scholar

[20]

N. J. A. Sloane and J. G. Thompson, Cyclic self-dual codes,, IEEE Trans. Inform. Theory, 29 (1983), 364.  doi: 10.1109/TIT.1983.1056682.  Google Scholar

show all references

References:
[1]

G. K. Bakshi and M. Raka, Self-dual and self-orthogonal negacyclic codes of length $2p^n$ over a finite field,, Finite Fields Appl., 19 (2013), 39.  doi: 10.1016/j.ffa.2012.10.003.  Google Scholar

[2]

D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes,, Appl. Algebra Engin. Commun. Comp., 18 (2007), 379.  doi: 10.1007/s00200-007-0043-z.  Google Scholar

[3]

D. Boucher and F. Ulmer, Self-dual skew codes and factorization of skew polynomials,, J. Symb. Comp., 60 (2014), 47.  doi: 10.1016/j.jsc.2013.10.003.  Google Scholar

[4]

X. Caruso and J. Leborgne, Some algorithms for skew polynomials over finite fields,, preprint, ().   Google Scholar

[5]

H. Q. Dinh, Repeated-root constacyclic codes of length $2 p^s$,, Finite Fields Appl., 18 (2012), 133.  doi: 10.1016/j.ffa.2011.07.003.  Google Scholar

[6]

J. von zur Gathen and J. Gerhard, Modern Computer Algebra,, Cambridge Univ. Press, (2013).  doi: 10.1017/CBO9781139856065.  Google Scholar

[7]

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

[8]

K. Guenda and T. A. Gulliver, Self-dual repeated root cyclic and negacyclic codes over finite fields,, in 2012 IEEE Int. Symp. Inform. Theory Proc., (2012), 2904.   Google Scholar

[9]

S. Han, J.-L. Kim, H. Lee and Y. Lee, Construction of quasi-cyclic self-dual codes,, Finite Fields Appl., 18 (2012), 613.  doi: 10.1016/j.ffa.2011.12.006.  Google Scholar

[10]

N. Jacobson, The Theory of Rings,, Amer. Math. Soc., (1943).   Google Scholar

[11]

S. Jia, S. Ling and C. Xing, On self-dual cyclic codes over finite fields,, IEEE Trans. Inform. Theory, 57 (2011), 2243.  doi: 10.1109/TIT.2010.2092415.  Google Scholar

[12]

X. Kai and S. Zhu, On cyclic self-dual codes,, Appl. Algebra Engin. Commun. Comp., 19 (2008), 509.  doi: 10.1007/s00200-008-0086-9.  Google Scholar

[13]

R. Lidl and H. Niederreiter, Finite Fields,, Cambridge Univ. Press, (1997).   Google Scholar

[14]

S. Ling, H. Niederreiter and P. Solé, On the algebraic structure of quasi-cyclic codes IV: Repeated roots Chain rings,, Des. Codes Crypt., 38 (2006), 337.  doi: 10.1007/s10623-005-1431-7.  Google Scholar

[15]

S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes I. Finite fields,, IEEE Trans. Inform. Theory, 47 (2001), 2751.  doi: 10.1109/18.959257.  Google Scholar

[16]

R. W. K. Odoni, On additive polynomials over a finite field,, Proc. Edinburgh Math. Soc., 42 (1999), 1.  doi: 10.1017/S0013091500019970.  Google Scholar

[17]

O. Ore, Theory of Non-Commutative Polynomials,, Ann. Math., 34 (1933), 480.  doi: 10.2307/1968173.  Google Scholar

[18]

A. Sahni and P. T. Sehgal, Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields,, Adv. Math. Commun., 9 (2015), 437.  doi: 10.3934/amc.2015.9.437.  Google Scholar

[19]

I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length,, Int. J. Inf. Coding Theory, 2 (2011), 10.  doi: 10.1504/IJICOT.2011.044674.  Google Scholar

[20]

N. J. A. Sloane and J. G. Thompson, Cyclic self-dual codes,, IEEE Trans. Inform. Theory, 29 (1983), 364.  doi: 10.1109/TIT.1983.1056682.  Google Scholar

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