August  2013, 7(3): 253-265. doi: 10.3934/amc.2013.7.253

On codes over rings invariant under affine groups

1. 

Institute of Mathematics, 125 Pushkin Str., Almaty 050010, Kazakhstan

Received  December 2011 Revised  March 2013 Published  July 2013

We give a description of extended cyclic codes of length $p^n$ over a field and over the ring of integers modulo $p^e$ admitting the affine group $AGL_m(p^t)$, $n=mt$, as a permutation group.
Citation: Kanat Abdukhalikov. On codes over rings invariant under affine groups. Advances in Mathematics of Communications, 2013, 7 (3) : 253-265. doi: 10.3934/amc.2013.7.253
References:
[1]

K. S. Abdukhalikov, Affine invariant and cyclic codes over $p$-adic numbers and finite rings, Des. Codes Cryptogr., 23 (2001), 343-370. doi: 10.1023/A:1011227228998.  Google Scholar

[2]

K. S. Abdukhalikov, Codes over $p$-adic numbers and finite rings invariant under the full affine group, Finite Fields Appl., 7 (2001), 449-467. doi: 10.1006/ffta.2000.0297.  Google Scholar

[3]

K. S. Abdukhalikov, Defining sets of cyclic codes invariant under the affine group, Electron. Notes Discrete Math., 6 (2001), 328-336. doi: 10.1016/S1571-0653(04)00184-2.  Google Scholar

[4]

K. S. Abdukhalikov, Lattices invariant under the affine general linear group, J. Algebra, 276 (2004), 638-662. doi: 10.1016/S0021-8693(03)00496-4.  Google Scholar

[5]

K. Abdukhalikov, Defining sets of extended cyclic codes invariant under the affine group, J. Pure Appl. Algebra, 196 (2005), 1-19. doi: 10.1016/j.jpaa.2004.02.016.  Google Scholar

[6]

K. Abdukhalikov, E. Bannai and S. Suda, Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets, J. Combin. Theory Ser. A, 116 (2009), 434-448. doi: 10.1016/j.jcta.2008.07.002.  Google Scholar

[7]

M. Bardoe and P. Sin, The permutation modules for $GL(n+1,\mathbb F_q)$ acting on $\mathbb P^n(\mathbb F_q)$ and $\mathbb F_q^{n+1}$, J. London Math. Soc. (2), 61 (2000), 58-80. doi: 10.1112/S002461079900839X.  Google Scholar

[8]

T. Berger and P. Charpin, The permutation group of affine-invariant extended cyclic codes, IEEE Trans. Inform. Theory, 42 (1996), 2194-2209. doi: 10.1109/18.556607.  Google Scholar

[9]

J. T. Blackford and D. K. Ray-Chaudhuri, A transform approach to permutation groups of cyclic codes over Galois rings, IEEE Trans. Inform. Theory, 46 (2000), 2050-2058. doi: 10.1109/18.887849.  Google Scholar

[10]

P. Delsarte, On cyclic codes that are invariant under the general linear group, IEEE Trans. Inform. Theory, 16 (1970), 760-769.  Google Scholar

[11]

B. K. Dey and B. S. Rajan, Affine invariant extended cyclic codes over Galois rings, IEEE Trans. Inform. Theory, 50 (2004), 691-698. doi: 10.1109/TIT.2004.825044.  Google Scholar

[12]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $Z_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.  Google Scholar

[13]

W. C. Huffman, Codes and groups, in "Handbook of Coding Theory" (eds. V.S. Pless and W.C. Huffman), Elsevier, 2 (1998), 1345-1440.  Google Scholar

[14]

T. Kasami, S. Lin and W. W. Peterson, Some results on cyclic codes which are invariant under the affine group and their applications, Inform. Control, 11 (1967), 475-496. doi: 10.1016/S0019-9958(67)90691-2.  Google Scholar

[15]

P. Sin, On codes that are invariant under the affine group, Electron. J. Combin., 19 (2012), 1-14.  Google Scholar

show all references

References:
[1]

K. S. Abdukhalikov, Affine invariant and cyclic codes over $p$-adic numbers and finite rings, Des. Codes Cryptogr., 23 (2001), 343-370. doi: 10.1023/A:1011227228998.  Google Scholar

[2]

K. S. Abdukhalikov, Codes over $p$-adic numbers and finite rings invariant under the full affine group, Finite Fields Appl., 7 (2001), 449-467. doi: 10.1006/ffta.2000.0297.  Google Scholar

[3]

K. S. Abdukhalikov, Defining sets of cyclic codes invariant under the affine group, Electron. Notes Discrete Math., 6 (2001), 328-336. doi: 10.1016/S1571-0653(04)00184-2.  Google Scholar

[4]

K. S. Abdukhalikov, Lattices invariant under the affine general linear group, J. Algebra, 276 (2004), 638-662. doi: 10.1016/S0021-8693(03)00496-4.  Google Scholar

[5]

K. Abdukhalikov, Defining sets of extended cyclic codes invariant under the affine group, J. Pure Appl. Algebra, 196 (2005), 1-19. doi: 10.1016/j.jpaa.2004.02.016.  Google Scholar

[6]

K. Abdukhalikov, E. Bannai and S. Suda, Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets, J. Combin. Theory Ser. A, 116 (2009), 434-448. doi: 10.1016/j.jcta.2008.07.002.  Google Scholar

[7]

M. Bardoe and P. Sin, The permutation modules for $GL(n+1,\mathbb F_q)$ acting on $\mathbb P^n(\mathbb F_q)$ and $\mathbb F_q^{n+1}$, J. London Math. Soc. (2), 61 (2000), 58-80. doi: 10.1112/S002461079900839X.  Google Scholar

[8]

T. Berger and P. Charpin, The permutation group of affine-invariant extended cyclic codes, IEEE Trans. Inform. Theory, 42 (1996), 2194-2209. doi: 10.1109/18.556607.  Google Scholar

[9]

J. T. Blackford and D. K. Ray-Chaudhuri, A transform approach to permutation groups of cyclic codes over Galois rings, IEEE Trans. Inform. Theory, 46 (2000), 2050-2058. doi: 10.1109/18.887849.  Google Scholar

[10]

P. Delsarte, On cyclic codes that are invariant under the general linear group, IEEE Trans. Inform. Theory, 16 (1970), 760-769.  Google Scholar

[11]

B. K. Dey and B. S. Rajan, Affine invariant extended cyclic codes over Galois rings, IEEE Trans. Inform. Theory, 50 (2004), 691-698. doi: 10.1109/TIT.2004.825044.  Google Scholar

[12]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $Z_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319. doi: 10.1109/18.312154.  Google Scholar

[13]

W. C. Huffman, Codes and groups, in "Handbook of Coding Theory" (eds. V.S. Pless and W.C. Huffman), Elsevier, 2 (1998), 1345-1440.  Google Scholar

[14]

T. Kasami, S. Lin and W. W. Peterson, Some results on cyclic codes which are invariant under the affine group and their applications, Inform. Control, 11 (1967), 475-496. doi: 10.1016/S0019-9958(67)90691-2.  Google Scholar

[15]

P. Sin, On codes that are invariant under the affine group, Electron. J. Combin., 19 (2012), 1-14.  Google Scholar

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