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Additive Toeplitz codes over $ \mathbb{F}_{4} $
1. | Department of Mathematics, Faculty of Science, Ankara University, Ankara, 06100, Turkey |
2. | Department of Mathematics, Faculty of Arts and Sciences, Nevşehir Hacı Bektaş Veli University, Nevşehir, 50300, Turkey |
In this paper, we introduce additive Toeplitz codes over $ \mathbb{F}_{4} $. The additive Toeplitz codes are a generalization of additive circulant codes over $ \mathbb{F}_{4} $. We find many optimal additive Toeplitz codes (OATC) over $ \mathbb{F}_{4} $. These optimal codes also contain optimal non-circulant codes, so we find new additive codes in this manner. We provide some theorems to partially classify OATC. Then, we give a new algorithm that fully classifies OATC by combining these theorems with Gaborit's algorithm. We classify OATC over $ \mathbb{F}_{4} $ of length up to $ 13 $. We obtain $ 2 $ inequivalent optimal additive toeplitz codes (IOATC) that are non-circulant codes of length $ 5 $, $ 92 $ of length $ 8 $, $ 2068 $ of length $ 9 $, and $ 39 $ of length $ 11 $. Moreover, we improve an idea related to quadratic residue codes to construct optimal and near-optimal additive Toeplitz codes over $ \mathbb{F}_{4} $ of length prime $ p $. We obtain many optimal and near-optimal additive Toeplitz codes for some primes $ p $ from this construction.
References:
[1] |
A. R. Calderbank, E. M. Rains, P. M. Shor and N. J. A. Sloane,
Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
doi: 10.1109/18.681315. |
[2] |
J. Cannon, W. Bosma, C. Fieker and A. Steel, Handbook of Magma Functions, Version 2.19, Sydney, 2013. Google Scholar |
[3] |
L. E. Danielsen and M. G. Parker,
Directed graph representation of half-rate additive codes over GF(4), Des. Codes Cryptogr., 59 (2011), 119-130.
doi: 10.1007/s10623-010-9469-6. |
[4] |
L. E. Danielsen and M. G. Parker,
On the classification of all self-dual additive codes over GF(4) of length up to 12, J. Combin. Theory Ser. A, 113 (2006), 1351-1367.
doi: 10.1016/j.jcta.2005.12.004. |
[5] |
P. Gaborit, W. C. Huffman, J. L. Kim and V. Pless, On additive GF(4) codes, DIMACS Workshop Codes Assoc. Schemes, DIMACS Ser. Discr. Math. Theoret. Comp. Sci., Amer. Math. Soc., 56 (2001), 135-149. Google Scholar |
[6] |
T. A. Gulliver and J.-L. Kim,
Circulant based extremal additive self-dual codes over GF(4), IEEE Trans. on Inform. Theory, 50 (2004), 359-366.
doi: 10.1109/TIT.2003.822616. |
[7] |
G. Höhn,
Self-dual codes over the Kleinian four group, Math. Ann., 327 (2003), 227-255.
doi: 10.1007/s00208-003-0440-y. |
[8] |
P. R. J. Östergard,
Classifying subspaces of Hamming spaces, Des. Codes Cryptogr., 27 (2002), 297-305.
doi: 10.1023/A:1019903407222. |
[9] |
V. S. Pless and W. C. Huffman, Handbook of Coding Theory, North-Holland, Amsterdam, 1998. Google Scholar |
[10] |
Z. Varbanov, Some new results for additive self-dual codes over GF(4), Serdica J. Comput., 1 (2007), 213-227. Google Scholar |
show all references
References:
[1] |
A. R. Calderbank, E. M. Rains, P. M. Shor and N. J. A. Sloane,
Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.
doi: 10.1109/18.681315. |
[2] |
J. Cannon, W. Bosma, C. Fieker and A. Steel, Handbook of Magma Functions, Version 2.19, Sydney, 2013. Google Scholar |
[3] |
L. E. Danielsen and M. G. Parker,
Directed graph representation of half-rate additive codes over GF(4), Des. Codes Cryptogr., 59 (2011), 119-130.
doi: 10.1007/s10623-010-9469-6. |
[4] |
L. E. Danielsen and M. G. Parker,
On the classification of all self-dual additive codes over GF(4) of length up to 12, J. Combin. Theory Ser. A, 113 (2006), 1351-1367.
doi: 10.1016/j.jcta.2005.12.004. |
[5] |
P. Gaborit, W. C. Huffman, J. L. Kim and V. Pless, On additive GF(4) codes, DIMACS Workshop Codes Assoc. Schemes, DIMACS Ser. Discr. Math. Theoret. Comp. Sci., Amer. Math. Soc., 56 (2001), 135-149. Google Scholar |
[6] |
T. A. Gulliver and J.-L. Kim,
Circulant based extremal additive self-dual codes over GF(4), IEEE Trans. on Inform. Theory, 50 (2004), 359-366.
doi: 10.1109/TIT.2003.822616. |
[7] |
G. Höhn,
Self-dual codes over the Kleinian four group, Math. Ann., 327 (2003), 227-255.
doi: 10.1007/s00208-003-0440-y. |
[8] |
P. R. J. Östergard,
Classifying subspaces of Hamming spaces, Des. Codes Cryptogr., 27 (2002), 297-305.
doi: 10.1023/A:1019903407222. |
[9] |
V. S. Pless and W. C. Huffman, Handbook of Coding Theory, North-Holland, Amsterdam, 1998. Google Scholar |
[10] |
Z. Varbanov, Some new results for additive self-dual codes over GF(4), Serdica J. Comput., 1 (2007), 213-227. Google Scholar |
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2018 Impact Factor: 0.879
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