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February  2018, 12(1): 189-198. doi: 10.3934/amc.2018013

## Long quasi-polycyclic $t-$ CIS codes

 1 Math. Dept., King Abdulaziz University, Jeddah, Saudi Arabia 2 Sabancı University, FENS, 34956 Istanbul, Turkey 3 Université de Paris 8, 2 rue de la Liberté, 93 526 Saint-Denis, France

Received  February 2017 Published  March 2018

We study complementary information set codes of length $tn$ and dimension $n$ of order $t$ called ($t-$CIS code for short). Quasi-cyclic (QC) and quasi-twisted (QT) $t$-CIS codes are enumerated by using their concatenated structure. Asymptotic existence results are derived for one-generator and fixed co-index QC and QT codes depending on Artin's primitive root conjecture. This shows that there are infinite families of rate $1/t$ long QC and QT $t$-CIS codes with relative distance satisfying a modified Varshamov-Gilbert bound. Similar results are defined for the new and more general class of quasi-polycyclic codes introduced recently by Berger and Amrani.

Citation: Adel Alahmadi, Cem Güneri, Hatoon Shoaib, Patrick Solé. Long quasi-polycyclic $t-$ CIS codes. Advances in Mathematics of Communications, 2018, 12 (1) : 189-198. doi: 10.3934/amc.2018013
##### References:
 [1] A. Alahmadi, C. Güneri, B. Özkaya, H. Shoaib and P. Solé, On self-dual double negacirculant codes, Discrete Appl. Math., 222 (2017), 205-212. [2] A. Alahmadi, S. T. Dougherty, A. Leroy and P. Solé, On the duality and the direction of polycyclic codes, Adv. Math. Commun., 10 (2016), 921--929. [3] N. Aydin and D. Ray-Chaudhuri, Quasi-cyclic codes over $\mathbb{Z}_4$ and some new binary codes, IEEE Trans. Inf. Theory, 48 (2002), 2065-2069. [4] T. P. Berger and N. E. Amrani, Codes over finite quotients of polynomial rings, Finite Fields Appl., 25 (2014), 165-181. [5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. Ⅰ. The user language, J. Symbolic Comput., 24 (1997), 235-265. [6] C. Carlet, F. Freibert, S. Guilley, M. Kiermaier, J.-L. Kim and P. Solé, Higher-order CIS codes, IEEE Trans. Inf. Theory, 60 (2014), 5283-5295. [7] C. Carlet, P. Gaborit, J.-L. Kim and P. Solé, A new class of codes for Boolean masking of cryptographic computations, IEEE Trans. Inf. Theory, 58 (2012), 6000-6011. [8] C. L. Chen, W. W. Peterson and E. J. Weldon, Some results on quasi-cyclic codes, Inf. Control, 15 (1969), 407-423. [9] M. Grassl, Tables of Linear Codes and Quantum Codes, available at www.codetables.de [10] A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $Z_4$ -linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inf. Theory,, 40 (1994), 301-319. [11] C. Hooley, On Artin's conjecture, J. Reine Angew. Math., 225 (1967), 209-220. [12] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, 2003. [13] Y. Jia, On quasi-twisted codes over finite fields, Finite Fields Appl., 18 (2012), 237-257. [14] R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Reading, 1983. [15] S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅰ: finite fields, IEEE Trans. Inf. Theory, 47 (2001), 2751-2760. [16] P. Moree, Artin's primitive root conjecture--a survey, Integers, 6 (2012), 1305-1416. [17] Z. -X. Wan, Quaternary Codes, WorldScientific, 1997.

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##### References:
 [1] A. Alahmadi, C. Güneri, B. Özkaya, H. Shoaib and P. Solé, On self-dual double negacirculant codes, Discrete Appl. Math., 222 (2017), 205-212. [2] A. Alahmadi, S. T. Dougherty, A. Leroy and P. Solé, On the duality and the direction of polycyclic codes, Adv. Math. Commun., 10 (2016), 921--929. [3] N. Aydin and D. Ray-Chaudhuri, Quasi-cyclic codes over $\mathbb{Z}_4$ and some new binary codes, IEEE Trans. Inf. Theory, 48 (2002), 2065-2069. [4] T. P. Berger and N. E. Amrani, Codes over finite quotients of polynomial rings, Finite Fields Appl., 25 (2014), 165-181. [5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. Ⅰ. The user language, J. Symbolic Comput., 24 (1997), 235-265. [6] C. Carlet, F. Freibert, S. Guilley, M. Kiermaier, J.-L. Kim and P. Solé, Higher-order CIS codes, IEEE Trans. Inf. Theory, 60 (2014), 5283-5295. [7] C. Carlet, P. Gaborit, J.-L. Kim and P. Solé, A new class of codes for Boolean masking of cryptographic computations, IEEE Trans. Inf. Theory, 58 (2012), 6000-6011. [8] C. L. Chen, W. W. Peterson and E. J. Weldon, Some results on quasi-cyclic codes, Inf. Control, 15 (1969), 407-423. [9] M. Grassl, Tables of Linear Codes and Quantum Codes, available at www.codetables.de [10] A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $Z_4$ -linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inf. Theory,, 40 (1994), 301-319. [11] C. Hooley, On Artin's conjecture, J. Reine Angew. Math., 225 (1967), 209-220. [12] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, 2003. [13] Y. Jia, On quasi-twisted codes over finite fields, Finite Fields Appl., 18 (2012), 237-257. [14] R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Reading, 1983. [15] S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅰ: finite fields, IEEE Trans. Inf. Theory, 47 (2001), 2751-2760. [16] P. Moree, Artin's primitive root conjecture--a survey, Integers, 6 (2012), 1305-1416. [17] Z. -X. Wan, Quaternary Codes, WorldScientific, 1997.
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