-
Previous Article
Dual-Ouroboros: An improvement of the McNie scheme
- AMC Home
- This Issue
-
Next Article
Complete weight enumerators of a class of linear codes over finite fields
On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $
Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain |
We introduce the Hadamard full propelinear codes that factorize as direct product of groups such that their associated group is $ C_{2t}\times C_2 $. We study the rank, the dimension of the kernel, and the structure of these codes. For several specific parameters we establish some links from circulant Hadamard matrices and the nonexistence of the codes we study. We prove that the dimension of the kernel of these codes is bounded by $ 3 $ if the code is nonlinear. We also get an equivalence between circulant complex Hadamard matrix and a type of Hadamard full propelinear code, and we find a new example of circulant complex Hadamard matrix of order $ 16 $.
References:
[1] |
V. Álvarez, F. Gudiel and M. B. Güemes,
On $\mathbb{Z}_t\times \mathbb{Z}_2^2$-cocyclic Hadamard matrices, J. Combin. Des., 23 (2015), 352-368.
doi: 10.1002/jcd.21406. |
[2] |
K. T. Arasu, W. de Launey and S. L. Ma,
On circulant complex Hadamard matrices, Des. Codes Cryptogr., 25 (2002), 123-142.
doi: 10.1023/A:1013817013980. |
[3] |
E. F. Assmus, Jr and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, 103. Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781316529836.![]() ![]() |
[4] |
I. Bailera, J. Borges and J. Rifà,
About some Hadamard full propelinear (2t, 2, 2)-codes. Rank and kernel, Electron. Notes Discret. Math., 54 (2016), 319-324.
doi: 10.1016/j.endm.2016.09.055. |
[5] |
A. Baliga and K. J. Horadam,
Cocyclic Hadamard matrices over $ {\mathbb{Z}}_t \times {\mathbb{Z}}_2^2$, Australas. J. Combin., 11 (1995), 123-134.
|
[6] |
S. Barrera Acevedo and H. Dietrich,
Perfect sequences over the quaternions and $(4n, 2, 4n, 2n)$-relative difference sets in $C_n \times Q_8$, Cryptogr. Commun., 10 (2018), 357-368.
doi: 10.1007/s12095-017-0224-y. |
[7] |
J. Borges, I. Y. Mogilnykh, J. Rifà and F. I. Solov'eva,
Structural properties of binary propelinear codes, Adv. Math. Commun., 6 (2012), 329-346.
doi: 10.3934/amc.2012.6.329. |
[8] |
W. Bosma, J. J. Cannon, C. Fieker and A. Steel, Handbook of Magma Functions, Edition 2.22, 2016. Google Scholar |
[9] |
A. T. Butson,
Relations among generalized Hadamard matrices, relative difference sets, and maximal length linear recurring sequences, Canad. J. Math., 15 (1963), 42-48.
doi: 10.4153/CJM-1963-005-3. |
[10] |
W. de Launey, D. L. Flannery and K. J. Horadam,
Cocyclic Hadamard matrices and difference sets, Discret. Appl. Math., 102 (2002), 47-61.
doi: 10.1016/S0166-218X(99)00230-9. |
[11] |
J. F. Dillon,
Some REALLY beautiful Hadamard matrices, Cryptogr. Commun., 2 (2010), 271-292.
doi: 10.1007/s12095-010-0031-1. |
[12] |
D. L. Flannery,
Cocyclic Hadamard matrices and Hadamard groups are equivalent, J. Algebra, 192 (1997), 749-779.
doi: 10.1006/jabr.1996.6949. |
[13] |
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() |
[14] |
N. Ito,
On Hadamard groups, J. Algebra, 168 (1994), 981-987.
doi: 10.1006/jabr.1994.1266. |
[15] |
N. Ito,
On Hadamard groups. II, J. Algebra, 169 (1994), 936-942.
doi: 10.1006/jabr.1994.1319. |
[16] |
N. Ito,
On Hadamard groups. III, Kyushu J. Math., 51 (1997), 369-379.
doi: 10.2206/kyushujm.51.369. |
[17] |
R. G. Kraemer,
Proof of a conjecture on Hadamard $2$-groups, J. Combin. Theory Ser. A, 63 (1993), 1-10.
doi: 10.1016/0097-3165(93)90012-W. |
[18] |
P. Ó Catháin and M. Röder,
The cocyclic Hadamard matrices of order less than 40, Des. Codes Cryptogr., 58 (2011), 73-88.
doi: 10.1007/s10623-010-9385-9. |
[19] |
K. T. Phelps, J. Rifà and M. Villanueva,
Hadamard codes of length $2^ts$ ($s$ odd). Rank and kernel, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Comput. Sci., Springer, Berlin, 3857 (2006), 328-337.
doi: 10.1007/11617983_32. |
[20] |
J. Rifà, Circulant Hadamard matrices as HFP-codes of type $C_4n\times C_2$, preprint, arXiv: 1711.09373v1. Google Scholar |
[21] |
J. Rifà, J. M. Basart and L. Huguet,
On completely regular propelinear codes, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. LNCS, Springer, Berlin, 357 (1989), 341-355.
doi: 10.1007/3-540-51083-4_71. |
[22] |
J. Rifà and E. Suárez Canedo,
Hadamard full propelinear codes of type Q. Rank and kernel, Des. Codes Cryptogr., 86 (2018), 1905-1921.
doi: 10.1007/s10623-017-0429-2. |
[23] |
J. Rifà i Coma and E. Suárez Canedo,
About a class of Hadamard propelinear codes, Conference on Discrete Mathematics and Computer Science, Electron. Notes Discret. Math. Elsevier Sci. B. V., Amsterdam, 46 (2014), 289-296.
doi: 10.1016/j.endm.2014.08.038. |
[24] |
H. J. Ryser, Combinatorial Mathematics, The Carus Mathematical Monographs, No. 14 Published by The Mathematical Association of America, New York, 1963.
doi: 10.1017/S0013091500011299. |
[25] |
B. Schmidt,
Williamson matrices and a conjecture of Ito's, Des. Codes Cryptogr., 17 (1999), 61-68.
doi: 10.1023/A:1008398319853. |
[26] |
R. J. Turyn,
Character sums and difference sets, Pacific J. Math., 15 (1965), 319-346.
doi: 10.2140/pjm.1965.15.319. |
show all references
References:
[1] |
V. Álvarez, F. Gudiel and M. B. Güemes,
On $\mathbb{Z}_t\times \mathbb{Z}_2^2$-cocyclic Hadamard matrices, J. Combin. Des., 23 (2015), 352-368.
doi: 10.1002/jcd.21406. |
[2] |
K. T. Arasu, W. de Launey and S. L. Ma,
On circulant complex Hadamard matrices, Des. Codes Cryptogr., 25 (2002), 123-142.
doi: 10.1023/A:1013817013980. |
[3] |
E. F. Assmus, Jr and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, 103. Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781316529836.![]() ![]() |
[4] |
I. Bailera, J. Borges and J. Rifà,
About some Hadamard full propelinear (2t, 2, 2)-codes. Rank and kernel, Electron. Notes Discret. Math., 54 (2016), 319-324.
doi: 10.1016/j.endm.2016.09.055. |
[5] |
A. Baliga and K. J. Horadam,
Cocyclic Hadamard matrices over $ {\mathbb{Z}}_t \times {\mathbb{Z}}_2^2$, Australas. J. Combin., 11 (1995), 123-134.
|
[6] |
S. Barrera Acevedo and H. Dietrich,
Perfect sequences over the quaternions and $(4n, 2, 4n, 2n)$-relative difference sets in $C_n \times Q_8$, Cryptogr. Commun., 10 (2018), 357-368.
doi: 10.1007/s12095-017-0224-y. |
[7] |
J. Borges, I. Y. Mogilnykh, J. Rifà and F. I. Solov'eva,
Structural properties of binary propelinear codes, Adv. Math. Commun., 6 (2012), 329-346.
doi: 10.3934/amc.2012.6.329. |
[8] |
W. Bosma, J. J. Cannon, C. Fieker and A. Steel, Handbook of Magma Functions, Edition 2.22, 2016. Google Scholar |
[9] |
A. T. Butson,
Relations among generalized Hadamard matrices, relative difference sets, and maximal length linear recurring sequences, Canad. J. Math., 15 (1963), 42-48.
doi: 10.4153/CJM-1963-005-3. |
[10] |
W. de Launey, D. L. Flannery and K. J. Horadam,
Cocyclic Hadamard matrices and difference sets, Discret. Appl. Math., 102 (2002), 47-61.
doi: 10.1016/S0166-218X(99)00230-9. |
[11] |
J. F. Dillon,
Some REALLY beautiful Hadamard matrices, Cryptogr. Commun., 2 (2010), 271-292.
doi: 10.1007/s12095-010-0031-1. |
[12] |
D. L. Flannery,
Cocyclic Hadamard matrices and Hadamard groups are equivalent, J. Algebra, 192 (1997), 749-779.
doi: 10.1006/jabr.1996.6949. |
[13] |
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() |
[14] |
N. Ito,
On Hadamard groups, J. Algebra, 168 (1994), 981-987.
doi: 10.1006/jabr.1994.1266. |
[15] |
N. Ito,
On Hadamard groups. II, J. Algebra, 169 (1994), 936-942.
doi: 10.1006/jabr.1994.1319. |
[16] |
N. Ito,
On Hadamard groups. III, Kyushu J. Math., 51 (1997), 369-379.
doi: 10.2206/kyushujm.51.369. |
[17] |
R. G. Kraemer,
Proof of a conjecture on Hadamard $2$-groups, J. Combin. Theory Ser. A, 63 (1993), 1-10.
doi: 10.1016/0097-3165(93)90012-W. |
[18] |
P. Ó Catháin and M. Röder,
The cocyclic Hadamard matrices of order less than 40, Des. Codes Cryptogr., 58 (2011), 73-88.
doi: 10.1007/s10623-010-9385-9. |
[19] |
K. T. Phelps, J. Rifà and M. Villanueva,
Hadamard codes of length $2^ts$ ($s$ odd). Rank and kernel, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Comput. Sci., Springer, Berlin, 3857 (2006), 328-337.
doi: 10.1007/11617983_32. |
[20] |
J. Rifà, Circulant Hadamard matrices as HFP-codes of type $C_4n\times C_2$, preprint, arXiv: 1711.09373v1. Google Scholar |
[21] |
J. Rifà, J. M. Basart and L. Huguet,
On completely regular propelinear codes, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. LNCS, Springer, Berlin, 357 (1989), 341-355.
doi: 10.1007/3-540-51083-4_71. |
[22] |
J. Rifà and E. Suárez Canedo,
Hadamard full propelinear codes of type Q. Rank and kernel, Des. Codes Cryptogr., 86 (2018), 1905-1921.
doi: 10.1007/s10623-017-0429-2. |
[23] |
J. Rifà i Coma and E. Suárez Canedo,
About a class of Hadamard propelinear codes, Conference on Discrete Mathematics and Computer Science, Electron. Notes Discret. Math. Elsevier Sci. B. V., Amsterdam, 46 (2014), 289-296.
doi: 10.1016/j.endm.2014.08.038. |
[24] |
H. J. Ryser, Combinatorial Mathematics, The Carus Mathematical Monographs, No. 14 Published by The Mathematical Association of America, New York, 1963.
doi: 10.1017/S0013091500011299. |
[25] |
B. Schmidt,
Williamson matrices and a conjecture of Ito's, Des. Codes Cryptogr., 17 (1999), 61-68.
doi: 10.1023/A:1008398319853. |
[26] |
R. J. Turyn,
Character sums and difference sets, Pacific J. Math., 15 (1965), 319-346.
doi: 10.2140/pjm.1965.15.319. |
even | |||
even square | |||
even | |||
odd |
even | |||
even square | |||
even | |||
odd |
t | ||||||||
1 | 3 | 3 | 3 | 3 | 3 | 3 | x | x |
2 | 4 | 4 | $\checkmark$ | $\checkmark$ | 4 | 4 | - | - |
3 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 11 | 1 |
4 | x | x | 5 | 5 | 7 | 2 | - | - |
6 | 3 | |||||||
5 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 19 | 1 |
6 | x | x | $\checkmark$ | $\checkmark$ | x | x | - | - |
7 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 27 | 1 |
8 | x | x | $\checkmark$ | $\checkmark$ | 11 | 2 | - | - |
13 | 1 | |||||||
9 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 35 | 1 |
10 | x | x | $\checkmark$ | $\checkmark$ | x | x | - | - |
t | ||||||||
1 | 3 | 3 | 3 | 3 | 3 | 3 | x | x |
2 | 4 | 4 | $\checkmark$ | $\checkmark$ | 4 | 4 | - | - |
3 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 11 | 1 |
4 | x | x | 5 | 5 | 7 | 2 | - | - |
6 | 3 | |||||||
5 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 19 | 1 |
6 | x | x | $\checkmark$ | $\checkmark$ | x | x | - | - |
7 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 27 | 1 |
8 | x | x | $\checkmark$ | $\checkmark$ | 11 | 2 | - | - |
13 | 1 | |||||||
9 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | 35 | 1 |
10 | x | x | $\checkmark$ | $\checkmark$ | x | x | - | - |
[1] |
Christos Koukouvinos, Dimitris E. Simos. Construction of new self-dual codes over $GF(5)$ using skew-Hadamard matrices. Advances in Mathematics of Communications, 2009, 3 (3) : 251-263. doi: 10.3934/amc.2009.3.251 |
[2] |
Ferenc Szöllősi. On quaternary complex Hadamard matrices of small orders. Advances in Mathematics of Communications, 2011, 5 (2) : 309-315. doi: 10.3934/amc.2011.5.309 |
[3] |
Gabriele Link. Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5577-5613. doi: 10.3934/dcds.2018245 |
[4] |
Olof Heden, Denis S. Krotov. On the structure of non-full-rank perfect $q$-ary codes. Advances in Mathematics of Communications, 2011, 5 (2) : 149-156. doi: 10.3934/amc.2011.5.149 |
[5] |
Joško Mandić, Tanja Vučičić. On the existence of Hadamard difference sets in groups of order 400. Advances in Mathematics of Communications, 2016, 10 (3) : 547-554. doi: 10.3934/amc.2016025 |
[6] |
Joaquim Borges, Ivan Yu. Mogilnykh, Josep Rifà, Faina I. Solov'eva. Structural properties of binary propelinear codes. Advances in Mathematics of Communications, 2012, 6 (3) : 329-346. doi: 10.3934/amc.2012.6.329 |
[7] |
Shu-Lin Lyu. On the Hermite--Hadamard inequality for convex functions of two variables. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 1-8. doi: 10.3934/naco.2014.4.1 |
[8] |
Francis C. Motta, Patrick D. Shipman. Informing the structure of complex Hadamard matrix spaces using a flow. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2349-2364. doi: 10.3934/dcdss.2019147 |
[9] |
Mariantonia Cotronei, Tomas Sauer. Full rank filters and polynomial reproduction. Communications on Pure & Applied Analysis, 2007, 6 (3) : 667-687. doi: 10.3934/cpaa.2007.6.667 |
[10] |
Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018 |
[11] |
S. S. Dragomir, I. Gomm. Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 271-278. doi: 10.3934/naco.2012.2.271 |
[12] |
Ruiyang Cai, Fudong Ge, Yangquan Chen, Chunhai Kou. Regional gradient controllability of ultra-slow diffusions involving the Hadamard-Caputo time fractional derivative. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019033 |
[13] |
Alessio Fiscella, Enzo Vitillaro. Local Hadamard well--posedness and blow--up for reaction--diffusion equations with non--linear dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5015-5047. doi: 10.3934/dcds.2013.33.5015 |
[14] |
Simon Foucart, Richard G. Lynch. Recovering low-rank matrices from binary measurements. Inverse Problems & Imaging, 2019, 13 (4) : 703-720. doi: 10.3934/ipi.2019032 |
[15] |
John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019 |
[16] |
Dean Crnković, Ronan Egan, Andrea Švob. Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020032 |
[17] |
Dean Crnković, Bernardo Gabriel Rodrigues, Sanja Rukavina, Loredana Simčić. Self-orthogonal codes from orbit matrices of 2-designs. Advances in Mathematics of Communications, 2013, 7 (2) : 161-174. doi: 10.3934/amc.2013.7.161 |
[18] |
Seungkook Park. Coherence of sensing matrices coming from algebraic-geometric codes. Advances in Mathematics of Communications, 2016, 10 (2) : 429-436. doi: 10.3934/amc.2016016 |
[19] |
Dina Ghinelli, Jennifer D. Key. Codes from incidence matrices and line graphs of Paley graphs. Advances in Mathematics of Communications, 2011, 5 (1) : 93-108. doi: 10.3934/amc.2011.5.93 |
[20] |
Behrouz Kheirfam. A full Nesterov-Todd step infeasible interior-point algorithm for symmetric optimization based on a specific kernel function. Numerical Algebra, Control & Optimization, 2013, 3 (4) : 601-614. doi: 10.3934/naco.2013.3.601 |
2018 Impact Factor: 0.879
Tools
Metrics
Other articles
by authors
[Back to Top]