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A new family of one-coincidence sets of sequences with dispersed elements for frequency hopping cdma systems
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 |
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.
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. |
show all references
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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