Note on the residue codes of self-dual \mathbb{Z}_4-codes having large minimum Lee weights

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November 2016, 10(4): 695-706. doi: 10.3934/amc.2016035

Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights

Masaaki Harada ^1,

1.	Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579

Received March 2014 Revised October 2014 Published November 2016

Abstract
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It is shown that the residue code of a self-dual $\mathbb{Z}_4$-code of length $24k$ (resp. $24k+8$) and minimum Lee weight $8k+4 \text{ or }8k+2$ (resp. $8k+8 \text{ or }8k+6$) is a binary extremal doubly even self-dual code for every positive integer $k$. A number of new self-dual $\mathbb{Z}_4$-codes of length $24$ and minimum Lee weight $10$ are constructed using the above characterization. These codes are Type I $\mathbb{Z}_4$-codes having the largest minimum Lee weight and the largest Euclidean weight among all Type I $\mathbb{Z}_4$-codes of that length. In addition, new extremal Type II $\mathbb{Z}_4$-codes of length $56$ are found.

Keywords: Self-dual code, doubly even code., binary code, Lee weight, $\mathbb{Z}_4$-code.

Mathematics Subject Classification: Primary: 94B05; Secondary: 94B2.

Citation: Masaaki Harada. Note on the residue codes of self-dual $\mathbb{Z}_4$-codes having large minimum Lee weights. Advances in Mathematics of Communications, 2016, 10 (4) : 695-706. doi: 10.3934/amc.2016035

References:

[1]	E. F. Assmus, Jr. and V. Pless, On the covering radius of extremal self-dual codes, IEEE Trans. Inform. Theory, 29 (1983), 359-363. doi: 10.1109/TIT.1983.1056681.
[2]	C. Bachoc and P. Gaborit, Designs and self-dual codes with long shadows, J. Combin. Theory Ser. A, 105 (2004), 15-34. doi: 10.1016/j.jcta.2003.09.003.
[3]	A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over $\mathbbZ_4$, IEEE Trans. Inform. Theory, 43 (1997), 969-976. doi: 10.1109/18.568705.
[4]	W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.
[5]	J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333. doi: 10.1109/18.59931.
[6]	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.
[7]	T. A. Gulliver and M. Harada, Certain self-dual codes over $\ZZ_4$ and the odd Leech lattice, in Int. Symp. Appl. Algebra Algebr. Algor. Error-Correcting Codes, Springer, Berlin, 1997, 130-137. doi: 10.1007/3-540-63163-1_10.
[8]	A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\ZZ_4$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.
[9]	M. Harada, Extremal type II $\mathbbZ_4$-codes of lengths $56$ and $64$, J. Combin. Theory Ser. A, 117 (2010), 1285-1288. doi: 10.1016/j.jcta.2009.09.003.
[10]	M. Kiermaier, There is no self-dual $\ZZ_4$-linear code whose Gray image has the parameters $(72,2^{36},16)$, IEEE Trans. Inform. Theory, 59 (2013), 3384-3386. doi: 10.1109/TIT.2013.2246816.
[11]	M. Kiermaier and A. Wassermann, Double and bordered $\alpha$-circulant self-dual codes over finite commutative chain rings, in Proc. 7th Int. Workshop Alg. Combin. Coding Theory, Pamporovo, 2008, 144-150.
[12]	M. Kiermaier and A. Wassermann, Minimum weights and weight enumerators of $\ZZ_4$-linear quadratic residue codes, IEEE Trans. Inform. Theory, 58 (2012), 4870-4883. doi: 10.1109/TIT.2012.2191389.
[13]	F. J. MacWilliams, N. J. A. Sloane and J. G. Thompson, Good self dual codes exist, Discrete Math., 3 (1972), 153-162.
[14]	C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200.
[15]	V. Pless, J. Leon and J. Fields, All $\ZZ_4$ codes of Type II and length 16 are known, J. Combin. Theory Ser. A, 78 (1997), 32-50. doi: 10.1006/jcta.1996.2750.
[16]	E. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.
[17]	E. Rains, Optimal self-dual codes over $\ZZ_4$, Discrete Math., 203 (1999), 215-228. doi: 10.1016/S0012-365X(98)00358-6.
[18]	E. Rains, Bounds for self-dual codes over $\ZZ_4$, Finite Fields Appl., 6 (2000), 146-163. doi: 10.1006/ffta.1999.0258.
[19]	E. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, 1998, 177-294.
[20]	E. Rains and N. J. A. Sloane, The shadow theory of modular and unimodular lattices, J. Number Theory, 73 (1998), 359-389. doi: 10.1006/jnth.1998.2306.
[21]	S. Zhang, On the nonexistence of extremal self-dual codes, Discrete Appl. Math., 91 (1999), 277-286. doi: 10.1016/S0166-218X(98)00131-0.

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References:

[1]	E. F. Assmus, Jr. and V. Pless, On the covering radius of extremal self-dual codes, IEEE Trans. Inform. Theory, 29 (1983), 359-363. doi: 10.1109/TIT.1983.1056681.
[2]	C. Bachoc and P. Gaborit, Designs and self-dual codes with long shadows, J. Combin. Theory Ser. A, 105 (2004), 15-34. doi: 10.1016/j.jcta.2003.09.003.
[3]	A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over $\mathbbZ_4$, IEEE Trans. Inform. Theory, 43 (1997), 969-976. doi: 10.1109/18.568705.
[4]	W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.
[5]	J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333. doi: 10.1109/18.59931.
[6]	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.
[7]	T. A. Gulliver and M. Harada, Certain self-dual codes over $\ZZ_4$ and the odd Leech lattice, in Int. Symp. Appl. Algebra Algebr. Algor. Error-Correcting Codes, Springer, Berlin, 1997, 130-137. doi: 10.1007/3-540-63163-1_10.
[8]	A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\ZZ_4$-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.
[9]	M. Harada, Extremal type II $\mathbbZ_4$-codes of lengths $56$ and $64$, J. Combin. Theory Ser. A, 117 (2010), 1285-1288. doi: 10.1016/j.jcta.2009.09.003.
[10]	M. Kiermaier, There is no self-dual $\ZZ_4$-linear code whose Gray image has the parameters $(72,2^{36},16)$, IEEE Trans. Inform. Theory, 59 (2013), 3384-3386. doi: 10.1109/TIT.2013.2246816.
[11]	M. Kiermaier and A. Wassermann, Double and bordered $\alpha$-circulant self-dual codes over finite commutative chain rings, in Proc. 7th Int. Workshop Alg. Combin. Coding Theory, Pamporovo, 2008, 144-150.
[12]	M. Kiermaier and A. Wassermann, Minimum weights and weight enumerators of $\ZZ_4$-linear quadratic residue codes, IEEE Trans. Inform. Theory, 58 (2012), 4870-4883. doi: 10.1109/TIT.2012.2191389.
[13]	F. J. MacWilliams, N. J. A. Sloane and J. G. Thompson, Good self dual codes exist, Discrete Math., 3 (1972), 153-162.
[14]	C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200.
[15]	V. Pless, J. Leon and J. Fields, All $\ZZ_4$ codes of Type II and length 16 are known, J. Combin. Theory Ser. A, 78 (1997), 32-50. doi: 10.1006/jcta.1996.2750.
[16]	E. Rains, Shadow bounds for self-dual codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.
[17]	E. Rains, Optimal self-dual codes over $\ZZ_4$, Discrete Math., 203 (1999), 215-228. doi: 10.1016/S0012-365X(98)00358-6.
[18]	E. Rains, Bounds for self-dual codes over $\ZZ_4$, Finite Fields Appl., 6 (2000), 146-163. doi: 10.1006/ffta.1999.0258.
[19]	E. Rains and N. J. A. Sloane, Self-dual codes, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, 1998, 177-294.
[20]	E. Rains and N. J. A. Sloane, The shadow theory of modular and unimodular lattices, J. Number Theory, 73 (1998), 359-389. doi: 10.1006/jnth.1998.2306.
[21]	S. Zhang, On the nonexistence of extremal self-dual codes, Discrete Appl. Math., 91 (1999), 277-286. doi: 10.1016/S0166-218X(98)00131-0.

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