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Empirical optimization of divisor arithmetic on hyperelliptic curves over $\mathbb{F}_{2^m}$
The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$
1. | Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano Bicocca, 20125 Milan, Italy, Italy |
2. | Lehrstuhl D für Mathematik, RWTH Aachen University, 52056 Aachen, Germany |
References:
[1] |
C. Aguilar Melchor and P. Gaborit, On the classification of extremal $[36,18,8]$ binary self-dual codes, IEEE Trans. Inform. Theory, 54 (2008), 4743-4750.
doi: 10.1109/TIT.2008.928976. |
[2] |
E. F. Assmuss and H. F. Mattson, New $5$-designs, J. Combin. Theory, 6 (1969), 122-151.
doi: 10.1016/S0021-9800(69)80115-8. |
[3] |
M. Borello, The automorphism group of a self-dual $[72,36,16]$ binary code does not contain elements of order $6$, IEEE Trans. Inform. Theory, 58 (2012), 7240-7245.
doi: 10.1109/TIT.2012.2211095. |
[4] |
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbol. Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[5] |
S. Bouyuklieva, On the automorphisms of order $2$ with fixed points for the extremal self-dual codes of length $24m$, Des. Codes Cryptogr., 25 (2002), 5-13.
doi: 10.1023/A:1012598832377. |
[6] |
S. Bouyuklieva, On the automorphism group of a doubly even $(72,36,16)$ code, IEEE Trans. Inform. Theory, 50 (2004), 544-547.
doi: 10.1109/TIT.2004.825252. |
[7] |
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, 112 (2006), 1351-1367.
doi: 10.1016/j.jcta.2005.12.004. |
[8] |
T. Feulner and G. Nebe, The automorphism group of an extremal $[72,36,16]$ code does not contain $Z_7$, $Z_3\times Z_3$, or $D_{10}$, IEEE Trans. Inform. Theory, 58 (2012), 6916-6924.
doi: 10.1109/TIT.2012.2208176. |
[9] |
W. C. Huffman, Automorphisms of codes with application to extremal doubly even codes of length $48$, IEEE Trans. Inform. Theory, IT-28 (1982), 511-521.
doi: 10.1109/TIT.1982.1056499. |
[10] |
C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200.
doi: 10.1016/S0019-9958(73)90273-8. |
[11] |
G. Nebe, An extremal $[72,36,16]$ binary code has no automorphism group containing $Z_2\times Z_4$, $Q_8$, or $Z_{10}$, Finite Fields Appl., 18 (2012), 563-566.
doi: 10.1016/j.ffa.2011.12.001. |
[12] |
E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139.
doi: 10.1109/18.651000. |
[13] |
N. J. A. Sloane, Is there a $(72; 36)$ $d = 16$ self-dual code?, IEEE Trans. Inform. Theory, 2 (1973), 251. |
show all references
References:
[1] |
C. Aguilar Melchor and P. Gaborit, On the classification of extremal $[36,18,8]$ binary self-dual codes, IEEE Trans. Inform. Theory, 54 (2008), 4743-4750.
doi: 10.1109/TIT.2008.928976. |
[2] |
E. F. Assmuss and H. F. Mattson, New $5$-designs, J. Combin. Theory, 6 (1969), 122-151.
doi: 10.1016/S0021-9800(69)80115-8. |
[3] |
M. Borello, The automorphism group of a self-dual $[72,36,16]$ binary code does not contain elements of order $6$, IEEE Trans. Inform. Theory, 58 (2012), 7240-7245.
doi: 10.1109/TIT.2012.2211095. |
[4] |
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symbol. Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
[5] |
S. Bouyuklieva, On the automorphisms of order $2$ with fixed points for the extremal self-dual codes of length $24m$, Des. Codes Cryptogr., 25 (2002), 5-13.
doi: 10.1023/A:1012598832377. |
[6] |
S. Bouyuklieva, On the automorphism group of a doubly even $(72,36,16)$ code, IEEE Trans. Inform. Theory, 50 (2004), 544-547.
doi: 10.1109/TIT.2004.825252. |
[7] |
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, 112 (2006), 1351-1367.
doi: 10.1016/j.jcta.2005.12.004. |
[8] |
T. Feulner and G. Nebe, The automorphism group of an extremal $[72,36,16]$ code does not contain $Z_7$, $Z_3\times Z_3$, or $D_{10}$, IEEE Trans. Inform. Theory, 58 (2012), 6916-6924.
doi: 10.1109/TIT.2012.2208176. |
[9] |
W. C. Huffman, Automorphisms of codes with application to extremal doubly even codes of length $48$, IEEE Trans. Inform. Theory, IT-28 (1982), 511-521.
doi: 10.1109/TIT.1982.1056499. |
[10] |
C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200.
doi: 10.1016/S0019-9958(73)90273-8. |
[11] |
G. Nebe, An extremal $[72,36,16]$ binary code has no automorphism group containing $Z_2\times Z_4$, $Q_8$, or $Z_{10}$, Finite Fields Appl., 18 (2012), 563-566.
doi: 10.1016/j.ffa.2011.12.001. |
[12] |
E. M. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139.
doi: 10.1109/18.651000. |
[13] |
N. J. A. Sloane, Is there a $(72; 36)$ $d = 16$ self-dual code?, IEEE Trans. Inform. Theory, 2 (1973), 251. |
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