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Some results on the structure of constacyclic codes and new linear codes over GF(7) from quasi-twisted codes
5-SEEDs from the lifted Golay code of length 24 over Z4
Institute of Computer Information Engineering, Jiangxi Normal University, Nanchang, Jiangxi 330022, China |
Spontaneous emission error designs (SEEDs) are combinatorial objects that can be used to construct quantum jump codes. The lifted Golay code $G_{24}$ of length $24$ over $\mathbb{Z}_4$ is cyclic self-dual code. It is known that $G_{24}$ yields $5$-designs. In this paper, by using the generator matrices of bordered double circulant codes, we obtain $22$ mutually disjoint $5$-$(24, k, \lambda)$ designs with $(k, \lambda)=(8, 1), $ $(10, 36), $ $(12,1584)$ and $5$-$(24, k;22)$-SEEDs for $k=8, $ $10, $ $12, $ $13$ from $G_{24}$.
References:
[1] |
M. Araya, M. Harada, V. D. Tonchev and A. Wassermann,
Mutually disjoint designs and new 5-designs derived from groups and codes, J. Combin. Des., 18 (2010), 305-317.
doi: 10.1002/jcd.20251. |
[2] |
T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado and M. Mussinger,
A new class of designs which protect against quantum jumps, Des.Codes Crypt., 29 (2003), 51-70.
doi: 10.1023/A:1024188005329. |
[3] |
A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain,
Type Ⅱ codes over $\mathbb{Z}_4$, IEEE Trans. Inform. Theory, 43 (1997), 969-976.
doi: 10.1109/18.568705. |
[4] |
A. Bonnecaze, P. Solé and A. R. Calderbank,
Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory, 41 (1995), 366-377.
doi: 10.1109/18.370138. |
[5] |
A. R. Calderbank and N. J. A. Sloane,
Modular and p-adic cyclic codes, Des. Codes Crypt., 6 (1995), 21-35.
doi: 10.1007/BF01390768. |
[6] |
J. Cannon and W. Bosma, Handbook of Magma functions, Version 2. 12, Univ. Sydney, 2005. |
[7] |
C. Charnes and T. Beth,
Combinatorial aspects of jump codes, Discrete Math., 294 (2005), 43-51.
doi: 10.1016/j.disc.2004.04.035. |
[8] |
J. H. Conway and N. J. A. Sloane,
Self-dual codes over the integers modulo 4, J. Combin. Theory Ser. A, 62 (1993), 30-45.
doi: 10.1016/0097-3165(93)90070-O. |
[9] |
J. Fang and Y. Chang,
Mutually disjoint t-designs and t-SEEDs from extremal doubly-even self-dual codes, Des. Codes Crypt., 73 (2014), 769-780.
doi: 10.1007/s10623-013-9825-4. |
[10] |
J. Fang and Y. Chang,
Mutually disjoint 5-designs and 5-spontaneous emission error designs from extremal ternary self-dual codes, J. Combin. Des., 23 (2015), 78-89.
doi: 10.1002/jcd.21391. |
[11] |
J. Fang, J. Zhou and Y. Chang,
Non-existence of some quantum jump codes with specified parameters, Des. Codes Crypt., 73 (2014), 223-235.
doi: 10.1007/s10623-013-9814-7. |
[12] |
T. A. Gulliver and M. Harada,
Extremal double circulant Type Ⅱ codes over $\mathbb{Z}_4$ and 5-(24. 10, 36) designs, Discrete Math., 194 (1999), 129-137.
doi: 10.1016/S0012-365X(98)00035-1. |
[13] |
A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé,
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. |
[14] |
M. Harada,
New 5-designs constructed from the lifted Golay code over $\mathbb{Z}_4$, J. Combin. Des., 6 (1999), 225-229.
doi: 10.1002/(SICI)1520-6610(1998)6:3<225::AID-JCD4>3.0.CO;2-H. |
[15] |
W. C. Huffman,
On the classification and enumeration of self-dual codes, Finite Fields Appl., 11 (2005), 451-490.
doi: 10.1016/j.ffa.2005.05.012. |
[16] |
W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() ![]() |
[17] | |
[18] |
M. Jimbo and K. Shiromoto,
A construction of mutually disjoint Steiner systems from isomorphic Golay codes, J. Combin. Theory Ser. A, 116 (2009), 1245-1251.
doi: 10.1016/j.jcta.2009.03.011. |
[19] |
M. Jimbo and K. Shiromoto,
Quantum jump codes and related combinatorial designs, Inf. Sec. Coding Theory Rel. Combin., 29 (2011), 285-311.
|
[20] |
V. Pless, Introduction to the Theory of Error-Correcting Codes, 3rd edition, Wiley, New York (1998).
doi: 10.1002/9781118032749. |
[21] |
V. Pless and Z. Qian,
Cyclic codes and quadratic residue codes over $\mathbb{Z}_4$, IEEE Trans. Inform. Theory, 42 (1996), 1594-1600.
doi: 10.1109/18.532906. |
[22] |
V. Pless, P. Solé and Z. Qian,
Cyclic self-dual $\mathbb{Z}_4$-codes, Finite Fields Appl., 3 (1997), 48-69.
doi: 10.1006/ffta.1996.0172. |
[23] |
K. Tanabe,
An Assmus-Mattson theorem for $\mathbb{Z}_4$-codes, IEEE Trans. Inform. Theory, 46 (2000), 48-53.
doi: 10.1109/18.817507. |
[24] |
J. V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1948. |
show all references
References:
[1] |
M. Araya, M. Harada, V. D. Tonchev and A. Wassermann,
Mutually disjoint designs and new 5-designs derived from groups and codes, J. Combin. Des., 18 (2010), 305-317.
doi: 10.1002/jcd.20251. |
[2] |
T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado and M. Mussinger,
A new class of designs which protect against quantum jumps, Des.Codes Crypt., 29 (2003), 51-70.
doi: 10.1023/A:1024188005329. |
[3] |
A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain,
Type Ⅱ codes over $\mathbb{Z}_4$, IEEE Trans. Inform. Theory, 43 (1997), 969-976.
doi: 10.1109/18.568705. |
[4] |
A. Bonnecaze, P. Solé and A. R. Calderbank,
Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory, 41 (1995), 366-377.
doi: 10.1109/18.370138. |
[5] |
A. R. Calderbank and N. J. A. Sloane,
Modular and p-adic cyclic codes, Des. Codes Crypt., 6 (1995), 21-35.
doi: 10.1007/BF01390768. |
[6] |
J. Cannon and W. Bosma, Handbook of Magma functions, Version 2. 12, Univ. Sydney, 2005. |
[7] |
C. Charnes and T. Beth,
Combinatorial aspects of jump codes, Discrete Math., 294 (2005), 43-51.
doi: 10.1016/j.disc.2004.04.035. |
[8] |
J. H. Conway and N. J. A. Sloane,
Self-dual codes over the integers modulo 4, J. Combin. Theory Ser. A, 62 (1993), 30-45.
doi: 10.1016/0097-3165(93)90070-O. |
[9] |
J. Fang and Y. Chang,
Mutually disjoint t-designs and t-SEEDs from extremal doubly-even self-dual codes, Des. Codes Crypt., 73 (2014), 769-780.
doi: 10.1007/s10623-013-9825-4. |
[10] |
J. Fang and Y. Chang,
Mutually disjoint 5-designs and 5-spontaneous emission error designs from extremal ternary self-dual codes, J. Combin. Des., 23 (2015), 78-89.
doi: 10.1002/jcd.21391. |
[11] |
J. Fang, J. Zhou and Y. Chang,
Non-existence of some quantum jump codes with specified parameters, Des. Codes Crypt., 73 (2014), 223-235.
doi: 10.1007/s10623-013-9814-7. |
[12] |
T. A. Gulliver and M. Harada,
Extremal double circulant Type Ⅱ codes over $\mathbb{Z}_4$ and 5-(24. 10, 36) designs, Discrete Math., 194 (1999), 129-137.
doi: 10.1016/S0012-365X(98)00035-1. |
[13] |
A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé,
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. |
[14] |
M. Harada,
New 5-designs constructed from the lifted Golay code over $\mathbb{Z}_4$, J. Combin. Des., 6 (1999), 225-229.
doi: 10.1002/(SICI)1520-6610(1998)6:3<225::AID-JCD4>3.0.CO;2-H. |
[15] |
W. C. Huffman,
On the classification and enumeration of self-dual codes, Finite Fields Appl., 11 (2005), 451-490.
doi: 10.1016/j.ffa.2005.05.012. |
[16] |
W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() ![]() |
[17] | |
[18] |
M. Jimbo and K. Shiromoto,
A construction of mutually disjoint Steiner systems from isomorphic Golay codes, J. Combin. Theory Ser. A, 116 (2009), 1245-1251.
doi: 10.1016/j.jcta.2009.03.011. |
[19] |
M. Jimbo and K. Shiromoto,
Quantum jump codes and related combinatorial designs, Inf. Sec. Coding Theory Rel. Combin., 29 (2011), 285-311.
|
[20] |
V. Pless, Introduction to the Theory of Error-Correcting Codes, 3rd edition, Wiley, New York (1998).
doi: 10.1002/9781118032749. |
[21] |
V. Pless and Z. Qian,
Cyclic codes and quadratic residue codes over $\mathbb{Z}_4$, IEEE Trans. Inform. Theory, 42 (1996), 1594-1600.
doi: 10.1109/18.532906. |
[22] |
V. Pless, P. Solé and Z. Qian,
Cyclic self-dual $\mathbb{Z}_4$-codes, Finite Fields Appl., 3 (1997), 48-69.
doi: 10.1006/ffta.1996.0172. |
[23] |
K. Tanabe,
An Assmus-Mattson theorem for $\mathbb{Z}_4$-codes, IEEE Trans. Inform. Theory, 46 (2000), 48-53.
doi: 10.1109/18.817507. |
[24] |
J. V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1948. |
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