5-SEEDs from the lifted Golay code of length 24 over Z<sub>4</sub>

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February 2017, 11(1): 259-266. doi: 10.3934/amc.2017017

5-SEEDs from the lifted Golay code of length 24 over Z₄

Institute of Computer Information Engineering, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

Received November 2015 Published February 2017

Fund Project: The author is supported by NSFC grant 11401271, 61462026, 61262015, 61462040.

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 $G_{24}$.

Keywords: Spontaneous emission error design (SEED), lifted Golay code, double circulant code, mutually disjoint t-design, cyclic code.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: Jianying Fang. 5-SEEDs from the lifted Golay code of length 24 over Z₄. Advances in Mathematics of Communications, 2017, 11 (1) : 259-266. doi: 10.3934/amc.2017017

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]	T. W. Hungerford, Algebra, Springer-Verlag, New York, 1974.
[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]	T. W. Hungerford, Algebra, Springer-Verlag, New York, 1974.
[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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