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 $G_{24}$.

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. ArayaM. HaradaV. 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.  Google Scholar

[2]

T. BethC. CharnesM. GrasslG. AlberA. 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.  Google Scholar

[3]

A. BonnecazeP. SoléC. Bachoc and B. Mourrain, Type Ⅱ codes over $\mathbb{Z}_4$, IEEE Trans. Inform. Theory, 43 (1997), 969-976.  doi: 10.1109/18.568705.  Google Scholar

[4]

A. BonnecazeP. 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.  Google Scholar

[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.  Google Scholar

[6]

J. Cannon and W. Bosma, Handbook of Magma functions, Version 2. 12, Univ. Sydney, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. FangJ. 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.  Google Scholar

[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.  Google Scholar

[13]

A. R. Hammons, Jr.P. V. KumarA. R. CalderbankN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[17]

T. W. Hungerford, Algebra, Springer-Verlag, New York, 1974.  Google Scholar

[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.  Google Scholar

[19]

M. Jimbo and K. Shiromoto, Quantum jump codes and related combinatorial designs, Inf. Sec. Coding Theory Rel. Combin., 29 (2011), 285-311.   Google Scholar

[20]

V. Pless, Introduction to the Theory of Error-Correcting Codes, 3rd edition, Wiley, New York (1998). doi: 10.1002/9781118032749.  Google Scholar

[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.  Google Scholar

[22]

V. PlessP. Solé and Z. Qian, Cyclic self-dual $\mathbb{Z}_4$-codes, Finite Fields Appl., 3 (1997), 48-69.  doi: 10.1006/ffta.1996.0172.  Google Scholar

[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.  Google Scholar

[24]

J. V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1948. Google Scholar

show all references

References:
[1]

M. ArayaM. HaradaV. 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.  Google Scholar

[2]

T. BethC. CharnesM. GrasslG. AlberA. 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.  Google Scholar

[3]

A. BonnecazeP. SoléC. Bachoc and B. Mourrain, Type Ⅱ codes over $\mathbb{Z}_4$, IEEE Trans. Inform. Theory, 43 (1997), 969-976.  doi: 10.1109/18.568705.  Google Scholar

[4]

A. BonnecazeP. 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.  Google Scholar

[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.  Google Scholar

[6]

J. Cannon and W. Bosma, Handbook of Magma functions, Version 2. 12, Univ. Sydney, 2005. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. FangJ. 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.  Google Scholar

[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.  Google Scholar

[13]

A. R. Hammons, Jr.P. V. KumarA. R. CalderbankN. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge Univ. Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[17]

T. W. Hungerford, Algebra, Springer-Verlag, New York, 1974.  Google Scholar

[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.  Google Scholar

[19]

M. Jimbo and K. Shiromoto, Quantum jump codes and related combinatorial designs, Inf. Sec. Coding Theory Rel. Combin., 29 (2011), 285-311.   Google Scholar

[20]

V. Pless, Introduction to the Theory of Error-Correcting Codes, 3rd edition, Wiley, New York (1998). doi: 10.1002/9781118032749.  Google Scholar

[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.  Google Scholar

[22]

V. PlessP. Solé and Z. Qian, Cyclic self-dual $\mathbb{Z}_4$-codes, Finite Fields Appl., 3 (1997), 48-69.  doi: 10.1006/ffta.1996.0172.  Google Scholar

[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.  Google Scholar

[24]

J. V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1948. Google Scholar

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