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

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

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.
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. doi: 10.1109/TIT.1983.1056681. Google Scholar

[2]

C. Bachoc and P. Gaborit, Designs and self-dual codes with long shadows,, J. Combin. Theory Ser. A, 105 (2004), 15. doi: 10.1016/j.jcta.2003.09.003. Google Scholar

[3]

A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over $\mathbbZ_4$,, IEEE Trans. Inform. Theory, 43 (1997), 969. doi: 10.1109/18.568705. Google Scholar

[4]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,, J. Symbolic Comput., 24 (1997), 235. doi: 10.1006/jsco.1996.0125. Google Scholar

[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. doi: 10.1109/18.59931. Google Scholar

[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. doi: 10.1016/0097-3165(93)90070-O. Google Scholar

[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, (1997), 130. doi: 10.1007/3-540-63163-1_10. Google Scholar

[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. doi: 10.1109/18.312154. Google Scholar

[9]

M. Harada, Extremal type II $\mathbbZ_4$-codes of lengths $56$ and $64$,, J. Combin. Theory Ser. A, 117 (2010), 1285. doi: 10.1016/j.jcta.2009.09.003. Google Scholar

[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. doi: 10.1109/TIT.2013.2246816. Google Scholar

[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, (2008), 144. Google Scholar

[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. doi: 10.1109/TIT.2012.2191389. Google Scholar

[13]

F. J. MacWilliams, N. J. A. Sloane and J. G. Thompson, Good self dual codes exist,, Discrete Math., 3 (1972), 153. Google Scholar

[14]

C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes,, Inform. Control, 22 (1973), 188. Google Scholar

[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. doi: 10.1006/jcta.1996.2750. Google Scholar

[16]

E. Rains, Shadow bounds for self-dual codes,, IEEE Trans. Inform. Theory, 44 (1998), 134. doi: 10.1109/18.651000. Google Scholar

[17]

E. Rains, Optimal self-dual codes over $\ZZ_4$,, Discrete Math., 203 (1999), 215. doi: 10.1016/S0012-365X(98)00358-6. Google Scholar

[18]

E. Rains, Bounds for self-dual codes over $\ZZ_4$,, Finite Fields Appl., 6 (2000), 146. doi: 10.1006/ffta.1999.0258. Google Scholar

[19]

E. Rains and N. J. A. Sloane, Self-dual codes,, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), (1998), 177. Google Scholar

[20]

E. Rains and N. J. A. Sloane, The shadow theory of modular and unimodular lattices,, J. Number Theory, 73 (1998), 359. doi: 10.1006/jnth.1998.2306. Google Scholar

[21]

S. Zhang, On the nonexistence of extremal self-dual codes,, Discrete Appl. Math., 91 (1999), 277. doi: 10.1016/S0166-218X(98)00131-0. Google Scholar

show all references

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. doi: 10.1109/TIT.1983.1056681. Google Scholar

[2]

C. Bachoc and P. Gaborit, Designs and self-dual codes with long shadows,, J. Combin. Theory Ser. A, 105 (2004), 15. doi: 10.1016/j.jcta.2003.09.003. Google Scholar

[3]

A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over $\mathbbZ_4$,, IEEE Trans. Inform. Theory, 43 (1997), 969. doi: 10.1109/18.568705. Google Scholar

[4]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language,, J. Symbolic Comput., 24 (1997), 235. doi: 10.1006/jsco.1996.0125. Google Scholar

[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. doi: 10.1109/18.59931. Google Scholar

[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. doi: 10.1016/0097-3165(93)90070-O. Google Scholar

[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, (1997), 130. doi: 10.1007/3-540-63163-1_10. Google Scholar

[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. doi: 10.1109/18.312154. Google Scholar

[9]

M. Harada, Extremal type II $\mathbbZ_4$-codes of lengths $56$ and $64$,, J. Combin. Theory Ser. A, 117 (2010), 1285. doi: 10.1016/j.jcta.2009.09.003. Google Scholar

[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. doi: 10.1109/TIT.2013.2246816. Google Scholar

[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, (2008), 144. Google Scholar

[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. doi: 10.1109/TIT.2012.2191389. Google Scholar

[13]

F. J. MacWilliams, N. J. A. Sloane and J. G. Thompson, Good self dual codes exist,, Discrete Math., 3 (1972), 153. Google Scholar

[14]

C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes,, Inform. Control, 22 (1973), 188. Google Scholar

[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. doi: 10.1006/jcta.1996.2750. Google Scholar

[16]

E. Rains, Shadow bounds for self-dual codes,, IEEE Trans. Inform. Theory, 44 (1998), 134. doi: 10.1109/18.651000. Google Scholar

[17]

E. Rains, Optimal self-dual codes over $\ZZ_4$,, Discrete Math., 203 (1999), 215. doi: 10.1016/S0012-365X(98)00358-6. Google Scholar

[18]

E. Rains, Bounds for self-dual codes over $\ZZ_4$,, Finite Fields Appl., 6 (2000), 146. doi: 10.1006/ffta.1999.0258. Google Scholar

[19]

E. Rains and N. J. A. Sloane, Self-dual codes,, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), (1998), 177. Google Scholar

[20]

E. Rains and N. J. A. Sloane, The shadow theory of modular and unimodular lattices,, J. Number Theory, 73 (1998), 359. doi: 10.1006/jnth.1998.2306. Google Scholar

[21]

S. Zhang, On the nonexistence of extremal self-dual codes,, Discrete Appl. Math., 91 (1999), 277. doi: 10.1016/S0166-218X(98)00131-0. Google Scholar

[1]

Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027

[2]

Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261

[3]

Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275

[4]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032

[5]

Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503

[6]

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

[7]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1

[8]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003

[9]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45

[10]

Masaaki Harada. New doubly even self-dual codes having minimum weight 20. Advances in Mathematics of Communications, 2020, 14 (1) : 89-96. doi: 10.3934/amc.2020007

[11]

Joaquim Borges, Steven T. Dougherty, Cristina Fernández-Córdoba. Characterization and constructions of self-dual codes over $\mathbb Z_2\times \mathbb Z_4$. Advances in Mathematics of Communications, 2012, 6 (3) : 287-303. doi: 10.3934/amc.2012.6.287

[12]

Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393

[13]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399

[14]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889

[15]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013

[16]

Masaaki Harada, Akihiro Munemasa. On the covering radii of extremal doubly even self-dual codes. Advances in Mathematics of Communications, 2007, 1 (2) : 251-256. doi: 10.3934/amc.2007.1.251

[17]

M. De Boeck, P. Vandendriessche. On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$. Advances in Mathematics of Communications, 2014, 8 (3) : 281-296. doi: 10.3934/amc.2014.8.281

[18]

Daniel Heinlein, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. A subspace code of size $ \bf{333} $ in the setting of a binary $ \bf{q} $-analog of the Fano plane. Advances in Mathematics of Communications, 2019, 13 (3) : 457-475. doi: 10.3934/amc.2019029

[19]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407

[20]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83

2018 Impact Factor: 0.879

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]