Geodesic
A note on regular subgroups of the automorphism group of the linear Hadamard code
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 1455-1462.

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We consider the subgroups of the automorphism group of the linear Hadamard code that act regularly on the codewords of the code. These subgroups correspond to the regular subgroups of the general affine group $\mathrm{GA}(r,2)$ with respect to the action on the vectors of $F_2^{r}$, where $n=2^r-1 $ is the length of the Hamadard code. We show that the dihedral group $D_{2^{r-1}}$ is a regular subgroup of $\mathrm{GA}(r,2)$ only when $r=3$. Following the approach of [13] we study the regular subgroups of the automorphism group of the Hamming code obtained from the regular subgroups of the automorphism group of the Hadamard code of length $15$.
Keywords: error-correcting code, regular action
Mots-clés : automorphism group, affine group. 
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I. Yu. Mogilnykh. A note on regular subgroups of the automorphism group of the linear Hadamard code. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 15 (2018), pp. 1455-1462. http://geodesic.mathdoc.fr/item/SEMR_2018_15_a80/

[1] E.R. Berlekamp, “Coding Theory and the Mathieu Groups”, Information and Control, 18 (1971), 40–64 | DOI | MR | Zbl

[2] A.A. Buturlakin, “Spectra of finite linear and unitary groups”, Algebra and Logic, 47 (2008), 91–99 | DOI | MR | Zbl

[3] J. Borges, J. Rifà, “A characterization of 1-perfect additive codes”, IEEE Transactions on Information Theory, 45 (1999), 1688–1697 | DOI | MR | Zbl

[4] J. Borges, I. Yu. Mogilnykh, J. Rifà, F.I. Solov'eva, “Structural properties of binary propelinear codes”, Advances in Mathematics of Communication, 6 (2012), 329–346 | DOI | MR | Zbl

[5] J. Borges, K.T. Phelps, J. Rifà, “The rank and kernel of extended 1-perfect Z4-linear and additive non-Z4-linear codes”, IEEE Transactions on Information Theory, 49 (2003), 2028–2034 | DOI | MR | Zbl

[6] A. Caranti, F. Dalla Volta, M. Sala, “Abelian regular subgroups of the affine group and radical rings”, Publicationes mathematicae Debrecen, 69 (2006), 297–308 | MR | Zbl

[7] M. Bellani, “Some remarks on regular subgroups of the affine group”, International Journal of Group Theory, 1 (2012), 17–23 | MR | Zbl

[8] P. Hegedus, “Regular subgroups of the Affine group”, Journal of Algebra, 225 (2000), 740–742 | DOI | MR | Zbl

[9] D.S. Krotov, M. Villanueva, “Classification of the Z2Z4-Linear Hadamard Codes and Their Automorphism Groups”, IEEE Transactions on Information Theory, 61 (2015), 887–894 | DOI | MR | Zbl

[10] D.S. Krotov, “Z4-linear Hadamard and extended perfect codes”, WCC2001, International Workshop on Coding and Cryptography, Electronic Notes in Discrete Mathematics, 6, 2001, 107–112 | DOI | MR | Zbl

[11] D. S. Krotov, “Z2k-Dual Binary Codes”, IEEE Transactions on Information Theory, 53 (2007), 1532–1537 | DOI | MR | Zbl

[12] F.J. MacWilliams, N. J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, 1977 | MR | Zbl

[13] I. Yu. Mogilnykh, “On extending propelinear structures of the Nordstrom-Robinson code to the Hamming code”, Problems of Information Transmission, 52 (2016), 289–298 | DOI | MR | Zbl

[14] K.T. Phelps, J. Rifà, “On binary 1-perfect additive codes: some structural properties”, IEEE Transactions on Information Theory, 48 (2002), 2587–2592 | DOI | MR | Zbl

[15] A. del Rio, J. Rifà, “Families of Hadamard Z2Z4Q8-codes”, IEEE Transactions on Information Theory, 59 (2013), 5140–5151 | DOI | MR | Zbl

[16] N.V. Semakov, V.A. Zinoviev, “Complete and Quasi-complete Balanced Codes”, Problems of Information Transmission, 5 (1969), 11–13 | MR | Zbl

[17] F.I. Solov'eva, S. Topalova, “On automorphism groups of perfect binary codes and Steiner triple systems”, Problems of Information Transmission, 36 (2000), 331–335 | MR | Zbl