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February  2021, 15(1): 35-54. doi: 10.3934/amc.2020041

On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $

Ivan Bailera , , Joaquim Borges and  Josep Rifà

Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

* Corresponding author: ivan.bailera@uab.cat

Received  February 2019 Revised  July 2019 Published  February 2021 Early access  November 2019

Fund Project: This work has been partially supported by the Spanish grant TIN2016-77918-P (AEI/FEDER, UE)

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 $.

Keywords: Hadamard full propelinear codes, Hadamard matrices, kernel, rank.
Mathematics Subject Classification: Primary: 5B, 5E, 94B.
Citation: Ivan Bailera, Joaquim Borges, Josep Rifà. On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $. Advances in Mathematics of Communications, 2021, 15 (1) : 35-54. doi: 10.3934/amc.2020041
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. F. Dillon, Some REALLY beautiful Hadamard matrices, Cryptogr. Commun., 2 (2010), 271-292.  doi: 10.1007/s12095-010-0031-1.  Google Scholar

[12]

D. L. Flannery, Cocyclic Hadamard matrices and Hadamard groups are equivalent, J. Algebra, 192 (1997), 749-779.  doi: 10.1006/jabr.1996.6949.  Google Scholar

[13] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[14]

N. Ito, On Hadamard groups, J. Algebra, 168 (1994), 981-987.  doi: 10.1006/jabr.1994.1266.  Google Scholar

[15]

N. Ito, On Hadamard groups. II, J. Algebra, 169 (1994), 936-942.  doi: 10.1006/jabr.1994.1319.  Google Scholar

[16]

N. Ito, On Hadamard groups. III, Kyushu J. Math., 51 (1997), 369-379.  doi: 10.2206/kyushujm.51.369.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

B. Schmidt, Williamson matrices and a conjecture of Ito's, Des. Codes Cryptogr., 17 (1999), 61-68.  doi: 10.1023/A:1008398319853.  Google Scholar

[26]

R. J. Turyn, Character sums and difference sets, Pacific J. Math., 15 (1965), 319-346.  doi: 10.2140/pjm.1965.15.319.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. F. Dillon, Some REALLY beautiful Hadamard matrices, Cryptogr. Commun., 2 (2010), 271-292.  doi: 10.1007/s12095-010-0031-1.  Google Scholar

[12]

D. L. Flannery, Cocyclic Hadamard matrices and Hadamard groups are equivalent, J. Algebra, 192 (1997), 749-779.  doi: 10.1006/jabr.1996.6949.  Google Scholar

[13] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[14]

N. Ito, On Hadamard groups, J. Algebra, 168 (1994), 981-987.  doi: 10.1006/jabr.1994.1266.  Google Scholar

[15]

N. Ito, On Hadamard groups. II, J. Algebra, 169 (1994), 936-942.  doi: 10.1006/jabr.1994.1319.  Google Scholar

[16]

N. Ito, On Hadamard groups. III, Kyushu J. Math., 51 (1997), 369-379.  doi: 10.2206/kyushujm.51.369.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

B. Schmidt, Williamson matrices and a conjecture of Ito's, Des. Codes Cryptogr., 17 (1999), 61-68.  doi: 10.1023/A:1008398319853.  Google Scholar

[26]

R. J. Turyn, Character sums and difference sets, Pacific J. Math., 15 (1965), 319-346.  doi: 10.2140/pjm.1965.15.319.  Google Scholar

Table 2.  Allowable values of the rank $ r $, and dimension of the kernel $ k $ for nonlinear $ \operatorname{HFP}(\cdot,\cdot,\cdot) $-codes of length $ 4t $
$ \operatorname{HFP}(\cdot,\cdot,\cdot) $ $ t $ $ r $ $ k $
$ (4t_ \mathbf{u},2) $ even $ \leq 2t $ $ 1 $
$ (2t,2,2_ \mathbf{u}) $ even square $ \leq 2t $ $ 1,2,3 $
$ (2t,4_ \mathbf{u}) $ even $ \leq 2t $ $ 1,2,3 $
$ (t,Q_ \mathbf{u}) $ odd $ 4t-1 $ $ 1 $
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$ \operatorname{HFP}(\cdot,\cdot,\cdot) $ $ t $ $ r $ $ k $
$ (4t_ \mathbf{u},2) $ even $ \leq 2t $ $ 1 $
$ (2t,2,2_ \mathbf{u}) $ even square $ \leq 2t $ $ 1,2,3 $
$ (2t,4_ \mathbf{u}) $ even $ \leq 2t $ $ 1,2,3 $
$ (t,Q_ \mathbf{u}) $ odd $ 4t-1 $ $ 1 $
Table 1.  Rank and dimension of the kernel of Hadamard full propelinear codes with associated group $ C_{2t} \times C_2 $. Symbol x means that the non-existence was checked with Magma by exhaustive search, symbol $\checkmark$ means that the non-existence was proved analytically, and "-" means that the code does not have $ C_{2t}\times C_2 $ as associated group. When the values for the rank and the dimension of the kernel appears in a box it means that they are the only values for that box
t $ (4t_ \mathbf{u},2) $ $ (2t,2,2_ \mathbf{u}) $ $ (2t,4_ \mathbf{u}) $ $ (t,Q_ \mathbf{u}) $
$ r $ $ k $ $ r $ $ k $ $ r $ $ k $ $ r $ $ k $
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 - -
Table Options

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t $ (4t_ \mathbf{u},2) $ $ (2t,2,2_ \mathbf{u}) $ $ (2t,4_ \mathbf{u}) $ $ (t,Q_ \mathbf{u}) $
$ r $ $ k $ $ r $ $ k $ $ r $ $ k $ $ r $ $ k $
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 - -
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