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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.

[2]

N. Aydin and T. Asamov, http://www.z4codes.info The database of Z4 codes (Accessed March, 2018).

[3]

N. Aydin and T. Asamov, A database of Z4 codes, J. Comb. Inf. Syst. Sci., 34 (2009), 1-12. 

[4]

M. Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.  doi: 10.1007/s10623-011-9494-0.

[5]

I. F. Blake, Codes over certain rings, Information and Control., 20 (1972), 396-404.  doi: 10.1016/S0019-9958(72)90223-9.

[6]

I. F. Blake, Codes over integer residue rings, Information and Control., 29 (1975), 295-300.  doi: 10.1016/S0019-9958(75)80001-5.

[7]

W. Bosma, J. J. Cannon, C. Fieker and A. Steel, Handbook of magma functions, Edition, 2 (2010), 5017 pages.

[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.

[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.

[10]

D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, In Proc. of 12th IMA International Conference, Cryptography and Coding, Cirencester, UK, LNCS, 5921 (2009), 38-55. doi: 10.1007/978-3-642-10868-6_3.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker Inc, New York, 1974.

[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.

[19]

E. Prange, Cyclic error-correcting codes in two symbols, Air Force Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN, (1957), 57-103. 

[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.

[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.

[22]

E. Spiegel, Codes over $\mathbb{Z}_m$, Information and Control., 35 (1977), 48-51.  doi: 10.1016/S0019-9958(77)90526-5.

[23]

E. Spiegel, Codes over $\mathbb{Z}_m$ (revisited), Information and Control., 37 (1978), 100-104.  doi: 10.1016/S0019-9958(78)90461-8.

[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.

[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.

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.

[2]

N. Aydin and T. Asamov, http://www.z4codes.info The database of Z4 codes (Accessed March, 2018).

[3]

N. Aydin and T. Asamov, A database of Z4 codes, J. Comb. Inf. Syst. Sci., 34 (2009), 1-12. 

[4]

M. Bhaintwal, Skew quasi-cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.  doi: 10.1007/s10623-011-9494-0.

[5]

I. F. Blake, Codes over certain rings, Information and Control., 20 (1972), 396-404.  doi: 10.1016/S0019-9958(72)90223-9.

[6]

I. F. Blake, Codes over integer residue rings, Information and Control., 29 (1975), 295-300.  doi: 10.1016/S0019-9958(75)80001-5.

[7]

W. Bosma, J. J. Cannon, C. Fieker and A. Steel, Handbook of magma functions, Edition, 2 (2010), 5017 pages.

[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.

[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.

[10]

D. Boucher and F. Ulmer, Codes as modules over skew polynomial rings, In Proc. of 12th IMA International Conference, Cryptography and Coding, Cirencester, UK, LNCS, 5921 (2009), 38-55. doi: 10.1007/978-3-642-10868-6_3.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

B. R. McDonald, Finite Rings with Identity, Marcel Dekker Inc, New York, 1974.

[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.

[19]

E. Prange, Cyclic error-correcting codes in two symbols, Air Force Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN, (1957), 57-103. 

[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.

[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.

[22]

E. Spiegel, Codes over $\mathbb{Z}_m$, Information and Control., 35 (1977), 48-51.  doi: 10.1016/S0019-9958(77)90526-5.

[23]

E. Spiegel, Codes over $\mathbb{Z}_m$ (revisited), Information and Control., 37 (1978), 100-104.  doi: 10.1016/S0019-9958(78)90461-8.

[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.

[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.

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

Download as excel

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

Download as excel

$C$ $\Phi(C)$ $Res(C)$ $C^*$
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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