A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation

Amit Sharma; Maheshanand Bhaintwal

doi:10.3934/amc.2018043

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November 2018, 12(4): 723-739. doi: 10.3934/amc.2018043

A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation

Amit Sharma ^, and Maheshanand Bhaintwal

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

* Corresponding author: Amit Sharma

Received October 2017 Revised March 2018 Published September 2018

In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over $R = \mathbb{Z}_4+u\mathbb{Z}_4;u^2 = 1$, with an automorphism $θ$ and a derivation $δ_θ$. We generalize the notion of cyclic codes to skew-cyclic codes with derivation, and call such codes as $δ_θ$-cyclic codes. Some properties of skew polynomial ring $R[x, θ, {δ_θ}]$ are presented. A $δ_θ$-cyclic code is proved to be a left $R[x, θ, {δ_θ}]$-submodule of $\frac{R[x, θ, {δ_θ}]}{\langle x^n-1 \rangle}$. The form of a parity-check matrix of a free $δ_θ$-cyclic codes of even length $n$ is presented. These codes are further generalized to double $δ_θ$-cyclic codes over $R$. We have obtained some new good codes over $\mathbb{Z}_4$ via Gray images and residue codes of these codes. The new codes obtained have been reported and added to the database of $\mathbb{Z}_4$-codes [2].

Keywords: Skew-cyclic codes, Gray map, automorphisms and derivations, factorization, double-cyclic codes.

Mathematics Subject Classification: Primary: 94B05, 94B15, 11T71; Secondary: 12D05.

Citation: Amit Sharma, Maheshanand Bhaintwal. A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation. Advances in Mathematics of Communications, 2018, 12 (4) : 723-739. doi: 10.3934/amc.2018043

References:

[1]	M. Araya, M. Harada, H. Ito and K. Saito, On the classification of Z4-codes, Adv. Math. Commun., 11 (2017), 747-756. doi: 10.3934/amc.2017054. Google Scholar
[2]	N. Aydin and T. Asamov, http://www.z4codes.info The database of Z4 codes (Accessed March, 2018). Google Scholar
[3]	N. Aydin and T. Asamov, A database of Z4 codes, J. Comb. Inf. Syst. Sci., 34 (2009), 1-12. Google Scholar
[4]	M. Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101. doi: 10.1007/s10623-011-9494-0. Google Scholar
[5]	I. F. Blake, Codes over certain rings, Information and Control., 20 (1972), 396-404. doi: 10.1016/S0019-9958(72)90223-9. Google Scholar
[6]	I. F. Blake, Codes over integer residue rings, Information and Control., 29 (1975), 295-300. doi: 10.1016/S0019-9958(75)80001-5. Google Scholar
[7]	W. Bosma, J. J. Cannon, C. Fieker and A. Steel, Handbook of magma functions, Edition, 2 (2010), 5017 pages. Google Scholar
[8]	D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. of Symbolic Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008. Google Scholar
[9]	D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z. Google Scholar
[10]	D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, In Proc. of 12^th IMA International Conference, Cryptography and Coding, Cirencester, UK, LNCS, 5921 (2009), 38-55. doi: 10.1007/978-3-642-10868-6_3. Google Scholar
[11]	D. Boucher, P. Sol$\acute{e}$ 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
[12]	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. Google Scholar
[13]	S. T. Dougherty and K. Shiromoto, Maximum distance codes over rings of order 4, IEEE Trans. Info Theory, 47 (2001), 400-404. doi: 10.1109/18.904544. Google Scholar
[14]	F. Gursoy, I. Siap and B. Yildiz, Construction of skew cyclic codes over $\mathbb{F}_q+v\mathbb{F}_q$, Adv. Math. Commum., 8 (2014), 313-322. doi: 10.3934/amc.2014.8.313. Google Scholar
[15]	Jr. A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. Sloane and P. Sol$\acute{e}$, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154. Google Scholar
[16]	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. Google Scholar
[17]	B. R. McDonald, Finite Rings with Identity, Marcel Dekker Inc, New York, 1974. Google Scholar
[18]	M. Ozen, F. Z. Uzekmek, N. Aydin and N. T. Ozzaim, Cyclic and some constacyclic codes over the ring $\frac{Z_4[u]}{\langle u^2-1\rangle}$, Finite Fields Appl., 38 (2016), 27-39. doi: 10.1016/j.ffa.2015.12.003. Google Scholar
[19]	E. Prange, Cyclic error-correcting codes in two symbols, Air Force Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN, (1957), 57-103. Google Scholar
[20]	M. Shi, L. Qian, L. Sok, N. Aydin and P. Sole, On constacyclic codes over $\frac{Z_4[u]}{\langle u^2-1 \rangle}$ and their Gray images, Finite Fields Appl., 45 (2017), 86-95. doi: 10.1016/j.ffa.2016.11.016. Google Scholar
[21]	I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory, 2 (2011), 10-20. doi: 10.1504/IJICOT.2011.044674. Google Scholar
[22]	E. Spiegel, Codes over $\mathbb{Z}_m$, Information and Control., 35 (1977), 48-51. doi: 10.1016/S0019-9958(77)90526-5. Google Scholar
[23]	E. Spiegel, Codes over $\mathbb{Z}_m$ (revisited), Information and Control., 37 (1978), 100-104. doi: 10.1016/S0019-9958(78)90461-8. Google Scholar
[24]	B. Yildiz and N. Aydin, On codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$ and their $\mathbb{Z}_4$-images, Int. J. Inf. Coding Theory, 2 (2014), 226-237. doi: 10.1504/IJICOT.2014.066107. Google Scholar
[25]	B. Yildiz and S. Karadeniz, Linear codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$: MacWilliams identities, projections, and formally self dual codes, Finite Fields Appl., 27 (2014), 24-40. doi: 10.1016/j.ffa.2013.12.007. Google Scholar

show all references

References:

[1]	M. Araya, M. Harada, H. Ito and K. Saito, On the classification of Z4-codes, Adv. Math. Commun., 11 (2017), 747-756. doi: 10.3934/amc.2017054. Google Scholar
[2]	N. Aydin and T. Asamov, http://www.z4codes.info The database of Z4 codes (Accessed March, 2018). Google Scholar
[3]	N. Aydin and T. Asamov, A database of Z4 codes, J. Comb. Inf. Syst. Sci., 34 (2009), 1-12. Google Scholar
[4]	M. Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101. doi: 10.1007/s10623-011-9494-0. Google Scholar
[5]	I. F. Blake, Codes over certain rings, Information and Control., 20 (1972), 396-404. doi: 10.1016/S0019-9958(72)90223-9. Google Scholar
[6]	I. F. Blake, Codes over integer residue rings, Information and Control., 29 (1975), 295-300. doi: 10.1016/S0019-9958(75)80001-5. Google Scholar
[7]	W. Bosma, J. J. Cannon, C. Fieker and A. Steel, Handbook of magma functions, Edition, 2 (2010), 5017 pages. Google Scholar
[8]	D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. of Symbolic Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008. Google Scholar
[9]	D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Engrg. Comm. Comput., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z. Google Scholar
[10]	D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, In Proc. of 12^th IMA International Conference, Cryptography and Coding, Cirencester, UK, LNCS, 5921 (2009), 38-55. doi: 10.1007/978-3-642-10868-6_3. Google Scholar
[11]	D. Boucher, P. Sol$\acute{e}$ 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
[12]	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. Google Scholar
[13]	S. T. Dougherty and K. Shiromoto, Maximum distance codes over rings of order 4, IEEE Trans. Info Theory, 47 (2001), 400-404. doi: 10.1109/18.904544. Google Scholar
[14]	F. Gursoy, I. Siap and B. Yildiz, Construction of skew cyclic codes over $\mathbb{F}_q+v\mathbb{F}_q$, Adv. Math. Commum., 8 (2014), 313-322. doi: 10.3934/amc.2014.8.313. Google Scholar
[15]	Jr. A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. Sloane and P. Sol$\acute{e}$, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154. Google Scholar
[16]	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. Google Scholar
[17]	B. R. McDonald, Finite Rings with Identity, Marcel Dekker Inc, New York, 1974. Google Scholar
[18]	M. Ozen, F. Z. Uzekmek, N. Aydin and N. T. Ozzaim, Cyclic and some constacyclic codes over the ring $\frac{Z_4[u]}{\langle u^2-1\rangle}$, Finite Fields Appl., 38 (2016), 27-39. doi: 10.1016/j.ffa.2015.12.003. Google Scholar
[19]	E. Prange, Cyclic error-correcting codes in two symbols, Air Force Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN, (1957), 57-103. Google Scholar
[20]	M. Shi, L. Qian, L. Sok, N. Aydin and P. Sole, On constacyclic codes over $\frac{Z_4[u]}{\langle u^2-1 \rangle}$ and their Gray images, Finite Fields Appl., 45 (2017), 86-95. doi: 10.1016/j.ffa.2016.11.016. Google Scholar
[21]	I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory, 2 (2011), 10-20. doi: 10.1504/IJICOT.2011.044674. Google Scholar
[22]	E. Spiegel, Codes over $\mathbb{Z}_m$, Information and Control., 35 (1977), 48-51. doi: 10.1016/S0019-9958(77)90526-5. Google Scholar
[23]	E. Spiegel, Codes over $\mathbb{Z}_m$ (revisited), Information and Control., 37 (1978), 100-104. doi: 10.1016/S0019-9958(78)90461-8. Google Scholar
[24]	B. Yildiz and N. Aydin, On codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$ and their $\mathbb{Z}_4$-images, Int. J. Inf. Coding Theory, 2 (2014), 226-237. doi: 10.1504/IJICOT.2014.066107. Google Scholar
[25]	B. Yildiz and S. Karadeniz, Linear codes over $\mathbb{Z}_4 + u\mathbb{Z}_4$: MacWilliams identities, projections, and formally self dual codes, Finite Fields Appl., 27 (2014), 24-40. doi: 10.1016/j.ffa.2013.12.007. Google Scholar

Figure 1.

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Table 1. *: = Existing good code [2,1], **: = New good code

	$C$	$\Phi(C)$	$Res(C)$	$C^*$
Set of generators	Code	$(n, 4^{k_1}2^{k_2}, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$
$\{g_1(x), xg_1(x), x^2g_1(x)\}$	$C_1$	${(10, 4^6, 2)^{}}$	${(5, 4^42^1, 2)^{*}}$	$\mathbf{(10, 4^82^2, 2)}^{**}$
$\{g_2(x), xg_2(x), x^2g_2(x)\}$	$C_2$	${(20, 4^6, 8)}$	$(10, 4^6, 4)^*$	$(20, 4^{12}, 4)^*$
$\{g_3(x), xg_3(x), x^2g_3(x)\}$	$C_3$	${(20, 4^6, 6)}$	$(10, 4^5, 6)^*$	$(20, 4^{10}, 6)$
$\left\{ {{g_4}\left( x \right),x{g_4}\left( x \right),{x^2}{g_4}\left( x \right),{x^3}{g_4}\left( x \right)} \right\}$	$C_4$	${(24, 4^8, 6)}$	$(12, 4^8, 4)^*$	$(24, 4^{16}, 4)^*$
$\left\{ {{g_5}\left( x \right),x{g_5}\left( x \right),{x^2}{g_5}\left( x \right),{x^3}{g_5}\left( x \right)} \right\}$	$C_5$	${(28, 4^8, 6)}$	$(14, 4^8, 5)^*$	$(28, 4^{16}, 5)^*$
$\left\{ {{g_6}\left( x \right),x{g_6}\left( x \right),{x^2}{g_6}\left( x \right),{x^3}{g_6}\left( x \right)} \right\}$	$C_6$	${(30, 4^8, 6)}$	$(15, 4^8, 6)^*$	$(30, 4^{16}, 6)$
$\left\{ {{g_7}\left( x \right),x{g_7}\left( x \right),{x^2}{g_7}\left( x \right),{x^3}{g_7}\left( x \right)} \right\}$	$C_7$	${(36, 4^8, 8)}$	$(18, 4^8, 8)^*$	$(36, 4^{16}, 8)^*$

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	$C$	$\Phi(C)$	$Res(C)$	$C^*$
Set of generators	Code	$(n, 4^{k_1}2^{k_2}, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$
$\{g_1(x), xg_1(x), x^2g_1(x)\}$	$C_1$	${(10, 4^6, 2)^{}}$	${(5, 4^42^1, 2)^{*}}$	$\mathbf{(10, 4^82^2, 2)}^{**}$
$\{g_2(x), xg_2(x), x^2g_2(x)\}$	$C_2$	${(20, 4^6, 8)}$	$(10, 4^6, 4)^*$	$(20, 4^{12}, 4)^*$
$\{g_3(x), xg_3(x), x^2g_3(x)\}$	$C_3$	${(20, 4^6, 6)}$	$(10, 4^5, 6)^*$	$(20, 4^{10}, 6)$
$\left\{ {{g_4}\left( x \right),x{g_4}\left( x \right),{x^2}{g_4}\left( x \right),{x^3}{g_4}\left( x \right)} \right\}$	$C_4$	${(24, 4^8, 6)}$	$(12, 4^8, 4)^*$	$(24, 4^{16}, 4)^*$
$\left\{ {{g_5}\left( x \right),x{g_5}\left( x \right),{x^2}{g_5}\left( x \right),{x^3}{g_5}\left( x \right)} \right\}$	$C_5$	${(28, 4^8, 6)}$	$(14, 4^8, 5)^*$	$(28, 4^{16}, 5)^*$
$\left\{ {{g_6}\left( x \right),x{g_6}\left( x \right),{x^2}{g_6}\left( x \right),{x^3}{g_6}\left( x \right)} \right\}$	$C_6$	${(30, 4^8, 6)}$	$(15, 4^8, 6)^*$	$(30, 4^{16}, 6)$
$\left\{ {{g_7}\left( x \right),x{g_7}\left( x \right),{x^2}{g_7}\left( x \right),{x^3}{g_7}\left( x \right)} \right\}$	$C_7$	${(36, 4^8, 8)}$	$(18, 4^8, 8)^*$	$(36, 4^{16}, 8)^*$

Table 2. $^*$: = Existing good code [2,1], $^{**}$: = New good code;

	$C$	$\Phi(C)$	$Res(C)$	$C^*$
Set of generators	Name	$(n, M, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$
$\{h_0(x), xh_1(x)\}$	$A_1$	${(10,128, 2)^{}}$	${(5, 4^32^1, 2)^{*}}$	$(10, 4^62^2, 2)$
$\{h_1(x), xh_1(x), x^2h_1(x)\}$	$A_2$	${(12, 4096, 2)^{}}$	${(6, 4^52^1, 2)^{*}}$	$\mathbf{(12, 4^{10}2^2, 2)}^{**}$
$\{h_2(x), xh_2(x), x^2h_2(x), x^3h_2(x)\}$	$A_3$	${(14, 65536, 2)^{}}$	${(7, 4^62^1, 2)^{*}}$	${(14, 4^{12}2^2, 2)}$
$\{h_3(x), xh_3(x), x^3h_2(x), x^3h_3(x)\}$	$A_3$	${(16, 65536, 4)^{}}$	$\mathbf{(8, 4^7, 2)^{**}}$	$\mathbf{(16, 4^{14}, 2)}^{**}$

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	$C$	$\Phi(C)$	$Res(C)$	$C^*$
Set of generators	Name	$(n, M, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$
$\{h_0(x), xh_1(x)\}$	$A_1$	${(10,128, 2)^{}}$	${(5, 4^32^1, 2)^{*}}$	$(10, 4^62^2, 2)$
$\{h_1(x), xh_1(x), x^2h_1(x)\}$	$A_2$	${(12, 4096, 2)^{}}$	${(6, 4^52^1, 2)^{*}}$	$\mathbf{(12, 4^{10}2^2, 2)}^{**}$
$\{h_2(x), xh_2(x), x^2h_2(x), x^3h_2(x)\}$	$A_3$	${(14, 65536, 2)^{}}$	${(7, 4^62^1, 2)^{*}}$	${(14, 4^{12}2^2, 2)}$
$\{h_3(x), xh_3(x), x^3h_2(x), x^3h_3(x)\}$	$A_3$	${(16, 65536, 4)^{}}$	$\mathbf{(8, 4^7, 2)^{**}}$	$\mathbf{(16, 4^{14}, 2)}^{**}$

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A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation

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