# American Institute of Mathematical Sciences

August  2017, 11(3): 615-634. doi: 10.3934/amc.2017046

## Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes

 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

Received  January 2016 Published  August 2017

Let $S$ be a unital ring, $S[t;\sigma,\delta]$ a skew polynomial ring where $\sigma$ is an injective endomorphism and $\delta$ a left $\sigma$-derivation, and suppose $f\in S[t;\sigma,\delta]$ has degree $m$ and an invertible leading coefficient. Using right division by $f$ to define the multiplication, we obtain unital nonassociative algebras $S_f$ on the set of skew polynomials in $S[t;\sigma,\delta]$ of degree less than $m$. We study the structure of these algebras.

When $S$ is a Galois ring and $f$ base irreducible, these algebras yield families of finite unital nonassociative rings $A$, whose set of (left or right) zero divisors has the form $pA$ for some prime $p$.

For reducible $f$, the $S_f$ can be employed both to design linear $(f,\sigma,\delta)$-codes over unital rings and to study their behaviour.

Citation: Susanne Pumplün. Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes. Advances in Mathematics of Communications, 2017, 11 (3) : 615-634. doi: 10.3934/amc.2017046
##### References:
 [1] Y. Alkhamees, The group of automorphisms of finite chain rings, Arab Gulf J. Scient. Res., 8 (1990), 17-28. [2] Y. Alkhamees, The determination of the group of automorphisms of a finite chain ring of characteristic p, Q. J. Math., 42 (1991), 387-391.  doi: 10.1093/qmath/42.1.387. [3] A. Batoul, K. Guenda and T. A. Gulliver, On self-dual cyclic codes over finite chain rings, Des. Codes Crypt., 70 (2014), 347-358.  doi: 10.1007/s10623-012-9696-0. [4] M. Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Crypt., 62 (2012), 85-101.  doi: 10.1007/s10623-011-9494-0. [5] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, AAECC, 18 (2007), 370-389.  doi: 10.1007/s00200-007-0043-z. [6] 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. [7] D. Boucher and F. Ulmer, Linear codes using skew polynomials with automorphisms and derivations, Des. Codes Cryptogr., 70 (2014), 405-431.  doi: 10.1007/s10623-012-9704-4. [8] M. Boulagouaz and A. Leroy, $(σ, δ)$-codes, Adv. Math. Commun., 7 (2013), 463-474.  doi: 10.3934/amc.2013.7.463. [9] C. Brown, Ph. D thesis, Univ. Nottingham, in preparation. [10] Y. Cao, On constacyclic codes over finite chain rings, Finite Fields Appl., 24 (2013), 124-135.  doi: 10.1016/j.ffa.2013.07.001. [11] P. M. Cohn, Skew Fields. Theory of General Division Rings Cambridge Univ. Press, Cambridge, 1995. doi: 10.1017/CBO9781139087193. [12] J. Ducoat and F. Oggier, Lattice encoding of cyclic codes from skew polynomial rings in Proc. 4th Int Castle Meet. Coding Theory Appl. Palmela, 2014. [13] J. Ducoat and F. Oggier, On skew polynomial codes and lattices from quotients of cyclic division algebras, Adv. Math. Commun., 10 (2016), 79-94.  doi: 10.3934/amc.2016.10.79. [14] C. Feng, R. W. Nobrega, F. R. Kschischang and D. Silva, Communication over finite-chain-ring matrix channels, IEEE Trans. Inf. Theory, 60 (2014), 5899-5917.  doi: 10.1109/TIT.2014.2346079. [15] N. Fogarty and H. Gluesing-Luerssen, A circulant approach to skew-constacyclic codes, Finite Fields Appl., 35 (2015), 92-114.  doi: 10.1016/j.ffa.2015.03.008. [16] J. Gao and Q. Kong, Qiong 1-generator quasi-cyclic codes over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+\dots+u^{s-1}\mathbb{F}_{p^m}$, J. Franklin Inst., 350 (2013), 3260-3276.  doi: 10.1016/j.jfranklin.2013.08.001. [17] M. Giesbrecht, Factoring in skew-polynomial rings over finite fields, J. Symb. Comput., 26 (1998), 463-486.  doi: 10.1006/jsco.1998.0224. [18] M. Giesbrecht and Y. Zhang, Factoring and decomposing Ore polynomials over $\mathbb F_q(t)$, Proc. 2003 Int. Symp. Symb. Alg. Comp. , ACM, New York, 2003,127–134. doi: 10.1145/860854.860888. [19] J. Gómez-Torrecillas, Basic module theory over non-commutative rings with computational aspects of operator algebras in Algebraic and Algorithmic Aspects of Differential and Integral Operators Springer, Berlin, 2012, 23-82. doi: 10.1007/978-3-642-54479-8_2. [20] J. Gómez-Torrecillas, F. J. Lobillo and G. Navarro, Factoring Ore polynomials over $\mathbb{F}_q(t)$ is difficult preprint, arXiv: 1505.07252 [21] S. González, V. T. Markov, C. Martíınez, A. A. Nechaev and I. F. Rúa, Nonassociative Galois rings (in Russian), Diskret. Mat. , 14 (2002), 117–132; translation in Discr. Math. Appl. , 12 (2002), 519–606. [22] S. González, V. T. Markov, C. Martínez, A. A. Nechaev and I. F. Rúa, On cyclic topassociative generalized Galois rings, in Finite Fields and Applications, Springer, Berlin, 2004, 25–39. doi: 10.1007/978-3-540-24633-6_3. [23] S. González, V. T. Markov, C. Martínez, A. A. Nechaev and I. F. Rúa, Cyclic generalized Galois rings, Comm. Algebra, 33 (2005), 4467-4478.  doi: 10.1080/00927870500274796. [24] S. González, C. Martínez, I. F. Rúa, V. T. Markov and A. A. Nechaev, Coordinate sets of generalized Galois rings, J. Algebra Appl., 3 (2004), 31-48.  doi: 10.1142/S0219498804000678. [25] N. Jacobson, Finite-Dimensional Division Algebras over Fields Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-02429-0. [26] S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 39-63.  doi: 10.3934/amc.2012.6.39. [27] B. Kong, X. Zheng and H. Ma, The depth spectrums of constacyclic codes over finite chain rings, Discrete Math., 338 (2015), 256-261.  doi: 10.1016/j.disc.2014.09.013. [28] M. Lavrauw and J. Sheekey, Semifields from skew polynomial rings, Adv. Geom., 13 (2013), 583-604.  doi: 10.1515/advgeom-2013-0003. [29] A. Leroy, Noncommutative polynomial maps J. Algebra Appl. 11 (2012), 16. doi: 10.1142/S0219498812500764. [30] X. Liu and H. Liu, LCD codes over finite chain rings, Finite Fields Appl., 34 (2015), 1-19.  doi: 10.1016/j.ffa.2015.01.004. [31] B. McDonald, Finite Rings with Identity Marcel Dekker Inc. , New York, 1974. [32] F. Oggier and B. A. Sethuraman, Quotients of orders in cyclic algebras and space-time codes, Adv. Math. Commun., 7 (2013), 441-461.  doi: 10.3934/amc.2013.7.441. [33] J.-C. Petit, Sur certains quasi-corps généralisant un type d'anneau-quotient, Sémin. Dubriel. Algébre Th. Nombr., 20 (1966), 1-18. [34] S. Pumplün, A note on linear codes and nonassociative algebras obtained from skew polynomial rings preprint, arXiv: 1504.00190 [35] S. Pumplün, How to obtain lattices from $(f, σ, δ)$-codes via a generalization of Construction A preprint, arXiv: 1607.03787 [36] S. Pumplün, Quotients of orders in algebras obtained from skew polynomials and possible applications preprint, arXiv: 1609.04201 [37] S. Pumplün, Tensor products of nonassociative cyclic algebras, J. Algebra, 451 (2016), 145-165.  doi: 10.1016/j.jalgebra.2015.12.007. [38] S. Pumplün and A. Steele, Classes of nonassociative algebras carrying maps of degree $n$ with interesting properties available at http://agt2.cie.uma.es/~loos/jordan/archive/semimult/semimult.pdf [39] S. Pumplün and A. Steele, Fast-decodable MIDO codes from nonassociative algebras, Int. J. Inf. Coding Theory, 3 (2015), 15-38.  doi: 10.1504/IJICOT.2015.068695. [40] S. Pumplün and A. Steele, The nonassociative algebras used to build fast-decodable space-time block codes, Adv. Math. Commun., 9 (2015), 449-469.  doi: 10.3934/amc.2015.9.449. [41] L. Rónyai, Factoring polynomials over finite fields, J. Algorithms, 9 (1988), 391-400.  doi: 10.1016/0196-6774(88)90029-6. [42] R. Sandler, Autotopism groups of some finite non-associative algebras, Amer. J. Math., 84 (1962), 239-264.  doi: 10.2307/2372761. [43] R. D. Schafer, An Introduction to Nonassociative Algebras Dover Publ. Inc. , New York, 1995. [44] M. F. Singer, Testing reducibility of linear differential operators: a group-theoretic perspective, Appl. Algebra Engrg. Comm. Comput., 7 (1996), 77-104.  doi: 10.1007/BF01191378. [45] A. Steele, Some New Classes of Algebras Ph. D thesis, Univ. Nottingham, 2013. [46] A. Steele, Nonassociative cyclic algebras, Israel J. Math., 200 (2014), 361-387.  doi: 10.1007/s11856-014-0021-7. [47] A. Steele, S. Pumplün and F. Oggier, MIDO space-time codes from associative and nonassociative cyclic algebras, Inf. Theory Workshop (ITW), IEEE, 2012,192–196. doi: 10.1007/s11856-014-0021-7. [48] E. A. Whelan, A note on finite local rings, Rocky Mount. J. Math., 22 (1992), 757-759.  doi: 10.1216/rmjm/1181072765.

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##### References:
 [1] Y. Alkhamees, The group of automorphisms of finite chain rings, Arab Gulf J. Scient. Res., 8 (1990), 17-28. [2] Y. Alkhamees, The determination of the group of automorphisms of a finite chain ring of characteristic p, Q. J. Math., 42 (1991), 387-391.  doi: 10.1093/qmath/42.1.387. [3] A. Batoul, K. Guenda and T. A. Gulliver, On self-dual cyclic codes over finite chain rings, Des. Codes Crypt., 70 (2014), 347-358.  doi: 10.1007/s10623-012-9696-0. [4] M. Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Crypt., 62 (2012), 85-101.  doi: 10.1007/s10623-011-9494-0. [5] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, AAECC, 18 (2007), 370-389.  doi: 10.1007/s00200-007-0043-z. [6] 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. [7] D. Boucher and F. Ulmer, Linear codes using skew polynomials with automorphisms and derivations, Des. Codes Cryptogr., 70 (2014), 405-431.  doi: 10.1007/s10623-012-9704-4. [8] M. Boulagouaz and A. Leroy, $(σ, δ)$-codes, Adv. Math. Commun., 7 (2013), 463-474.  doi: 10.3934/amc.2013.7.463. [9] C. Brown, Ph. D thesis, Univ. Nottingham, in preparation. [10] Y. Cao, On constacyclic codes over finite chain rings, Finite Fields Appl., 24 (2013), 124-135.  doi: 10.1016/j.ffa.2013.07.001. [11] P. M. Cohn, Skew Fields. Theory of General Division Rings Cambridge Univ. Press, Cambridge, 1995. doi: 10.1017/CBO9781139087193. [12] J. Ducoat and F. Oggier, Lattice encoding of cyclic codes from skew polynomial rings in Proc. 4th Int Castle Meet. Coding Theory Appl. Palmela, 2014. [13] J. Ducoat and F. Oggier, On skew polynomial codes and lattices from quotients of cyclic division algebras, Adv. Math. Commun., 10 (2016), 79-94.  doi: 10.3934/amc.2016.10.79. [14] C. Feng, R. W. Nobrega, F. R. Kschischang and D. Silva, Communication over finite-chain-ring matrix channels, IEEE Trans. Inf. Theory, 60 (2014), 5899-5917.  doi: 10.1109/TIT.2014.2346079. [15] N. Fogarty and H. Gluesing-Luerssen, A circulant approach to skew-constacyclic codes, Finite Fields Appl., 35 (2015), 92-114.  doi: 10.1016/j.ffa.2015.03.008. [16] J. Gao and Q. Kong, Qiong 1-generator quasi-cyclic codes over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+\dots+u^{s-1}\mathbb{F}_{p^m}$, J. Franklin Inst., 350 (2013), 3260-3276.  doi: 10.1016/j.jfranklin.2013.08.001. [17] M. Giesbrecht, Factoring in skew-polynomial rings over finite fields, J. Symb. Comput., 26 (1998), 463-486.  doi: 10.1006/jsco.1998.0224. [18] M. Giesbrecht and Y. Zhang, Factoring and decomposing Ore polynomials over $\mathbb F_q(t)$, Proc. 2003 Int. Symp. Symb. Alg. Comp. , ACM, New York, 2003,127–134. doi: 10.1145/860854.860888. [19] J. Gómez-Torrecillas, Basic module theory over non-commutative rings with computational aspects of operator algebras in Algebraic and Algorithmic Aspects of Differential and Integral Operators Springer, Berlin, 2012, 23-82. doi: 10.1007/978-3-642-54479-8_2. [20] J. Gómez-Torrecillas, F. J. Lobillo and G. Navarro, Factoring Ore polynomials over $\mathbb{F}_q(t)$ is difficult preprint, arXiv: 1505.07252 [21] S. González, V. T. Markov, C. Martíınez, A. A. Nechaev and I. F. Rúa, Nonassociative Galois rings (in Russian), Diskret. Mat. , 14 (2002), 117–132; translation in Discr. Math. Appl. , 12 (2002), 519–606. [22] S. González, V. T. Markov, C. Martínez, A. A. Nechaev and I. F. Rúa, On cyclic topassociative generalized Galois rings, in Finite Fields and Applications, Springer, Berlin, 2004, 25–39. doi: 10.1007/978-3-540-24633-6_3. [23] S. González, V. T. Markov, C. Martínez, A. A. Nechaev and I. F. Rúa, Cyclic generalized Galois rings, Comm. Algebra, 33 (2005), 4467-4478.  doi: 10.1080/00927870500274796. [24] S. González, C. Martínez, I. F. Rúa, V. T. Markov and A. A. Nechaev, Coordinate sets of generalized Galois rings, J. Algebra Appl., 3 (2004), 31-48.  doi: 10.1142/S0219498804000678. [25] N. Jacobson, Finite-Dimensional Division Algebras over Fields Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-642-02429-0. [26] S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun., 6 (2012), 39-63.  doi: 10.3934/amc.2012.6.39. [27] B. Kong, X. Zheng and H. Ma, The depth spectrums of constacyclic codes over finite chain rings, Discrete Math., 338 (2015), 256-261.  doi: 10.1016/j.disc.2014.09.013. [28] M. Lavrauw and J. Sheekey, Semifields from skew polynomial rings, Adv. Geom., 13 (2013), 583-604.  doi: 10.1515/advgeom-2013-0003. [29] A. Leroy, Noncommutative polynomial maps J. Algebra Appl. 11 (2012), 16. doi: 10.1142/S0219498812500764. [30] X. Liu and H. Liu, LCD codes over finite chain rings, Finite Fields Appl., 34 (2015), 1-19.  doi: 10.1016/j.ffa.2015.01.004. [31] B. McDonald, Finite Rings with Identity Marcel Dekker Inc. , New York, 1974. [32] F. Oggier and B. A. Sethuraman, Quotients of orders in cyclic algebras and space-time codes, Adv. Math. Commun., 7 (2013), 441-461.  doi: 10.3934/amc.2013.7.441. [33] J.-C. Petit, Sur certains quasi-corps généralisant un type d'anneau-quotient, Sémin. Dubriel. Algébre Th. Nombr., 20 (1966), 1-18. [34] S. Pumplün, A note on linear codes and nonassociative algebras obtained from skew polynomial rings preprint, arXiv: 1504.00190 [35] S. Pumplün, How to obtain lattices from $(f, σ, δ)$-codes via a generalization of Construction A preprint, arXiv: 1607.03787 [36] S. Pumplün, Quotients of orders in algebras obtained from skew polynomials and possible applications preprint, arXiv: 1609.04201 [37] S. Pumplün, Tensor products of nonassociative cyclic algebras, J. Algebra, 451 (2016), 145-165.  doi: 10.1016/j.jalgebra.2015.12.007. [38] S. Pumplün and A. Steele, Classes of nonassociative algebras carrying maps of degree $n$ with interesting properties available at http://agt2.cie.uma.es/~loos/jordan/archive/semimult/semimult.pdf [39] S. Pumplün and A. Steele, Fast-decodable MIDO codes from nonassociative algebras, Int. J. Inf. Coding Theory, 3 (2015), 15-38.  doi: 10.1504/IJICOT.2015.068695. [40] S. Pumplün and A. Steele, The nonassociative algebras used to build fast-decodable space-time block codes, Adv. Math. Commun., 9 (2015), 449-469.  doi: 10.3934/amc.2015.9.449. [41] L. Rónyai, Factoring polynomials over finite fields, J. Algorithms, 9 (1988), 391-400.  doi: 10.1016/0196-6774(88)90029-6. [42] R. Sandler, Autotopism groups of some finite non-associative algebras, Amer. J. Math., 84 (1962), 239-264.  doi: 10.2307/2372761. [43] R. D. Schafer, An Introduction to Nonassociative Algebras Dover Publ. Inc. , New York, 1995. [44] M. F. Singer, Testing reducibility of linear differential operators: a group-theoretic perspective, Appl. Algebra Engrg. Comm. Comput., 7 (1996), 77-104.  doi: 10.1007/BF01191378. [45] A. Steele, Some New Classes of Algebras Ph. D thesis, Univ. Nottingham, 2013. [46] A. Steele, Nonassociative cyclic algebras, Israel J. Math., 200 (2014), 361-387.  doi: 10.1007/s11856-014-0021-7. [47] A. Steele, S. Pumplün and F. Oggier, MIDO space-time codes from associative and nonassociative cyclic algebras, Inf. Theory Workshop (ITW), IEEE, 2012,192–196. doi: 10.1007/s11856-014-0021-7. [48] E. A. Whelan, A note on finite local rings, Rocky Mount. J. Math., 22 (1992), 757-759.  doi: 10.1216/rmjm/1181072765.
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