May  2011, 5(2): 339-350. doi: 10.3934/amc.2011.5.339

Some optimal codes related to graphs invariant under the alternating group $A_8$

1. 

School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041, South Africa

Received  May 2010 Revised  March 2011 Published  May 2011

The alternating group $A_8$, acts as a primitive rank-3 group of degree $35$ on the set of lines of $V_4(2)$ with line stabilizer isomorphic to $2^4:(S_3 \times S_3)$ and orbits of lengths 1, 16 and 18 respectively. This action defines the unique strongly regular $(35, 16, 6, 8)$ graph. The paper examines the binary (resp. ternary) codes spanned by the rows of this graph, and its complement. We establish some properties of the codes and use the geometry of the designs and graphs to give an account on the nature of some classes of codewords, in particular those of minimum weight. Further, we show that the codes with parameters $[35, 28, 4]_2,[35, 6, 16]_2,[35, 29, 3]_2,[28, 7, 12]_2,[28, 21,4]_2, [36, 7, 16]_2, [36,29,4]_2$ and $[64, 56, 4]_2$ are all optimal. In addition, we show that the codes with parameters $[35, 13, 12]_3, [35, 22, 5]_3,[35, 14, 11]_3, [35, 21, 6]_3$ are all near-optimal for the given length and dimension.
Citation: Bernardo Gabriel Rodrigues. Some optimal codes related to graphs invariant under the alternating group $A_8$. Advances in Mathematics of Communications, 2011, 5 (2) : 339-350. doi: 10.3934/amc.2011.5.339
References:
[1]

E. F. Assmus, Jr. and J. D. Key, "Designs and Their Codes,'' Cambridge University Press, 1992.

[2]

E. F. Assmus, Jr. and J. D. Key, Hadamard matrices and their designs: a coding-theoretic approach, Trans. Amer. Math. Soc., 330 (1992), 269-293. doi: 10.2307/2154164.

[3]

W. Bosma and J. Cannon, Handbook of Magma Functions, Department of Mathematics, University of Sydney, November 1994, available online at http://magma.maths.usyd.edu.au/magma

[4]

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122. doi: 10.1112/blms/18.2.97.

[5]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, "Atlas of Finite Groups,'' Oxford University Press, Oxford, 1985.

[6]

D. Crnković, V. Mikulić and S. Rukavina, Block designs constructed from the group U$(3,3)$, J. Appl. Algebra Discrete Struct., 2 (2004), 69-81.

[7]

U. Dempwolff, Primitive rank-$3$ groups on symmetric designs, Des. Codes Crypt., 22 (2001), 191-207. doi: 10.1023/A:1008373207617.

[8]

L. E. Dickson, "Linear Groups with an Exposition of the Galois Field Theory,'' Dover Publications, New York, 1958.

[9]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, available online at http://www.codetables.de

[10]

W. H. Haemers, R. Peeters and J. M. van Rijckevorsel, Binary codes of strongly regular graphs, Des. Codes Crypt., 17 (1999), 187-209. doi: 10.1023/A:1026479210284.

[11]

C. Jansen, K. Lux, R. Parker and R. Wilson, "An Atlas of Brauer Characters,'' with Appendix 2 by T. Breuer and S. Norton, The Clarendon Press, Oxford University Press, New York, 1995.

[12]

W. M. Kantor, Symplectic groups, symmetric designs, and line ovals, J. Algebra, 33 (1975), 43-58. doi: 10.1016/0021-8693(75)90130-1.

[13]

J. D. Key and J. Moori, Codes, designs and graphs from the Janko groups $J_1$ and $J_2$, J. Combin. Math. Combin. Comput., 40 (2002), 143-159.

[14]

J. D. Key and J. Moori, Correction to: Codes, designs and graphs from the Janko groups $J_1$ and $J_2$, [J. Combin. Math. Combin. Comput., 40 (2002), 143-159], J. Combin. Math. Combin. Comput., 64 (2008), 153.

[15]

J. D. Key, J. Moori and B. G. Rodrigues, On some designs and codes from primitive representations of some finite simple groups, J. Combin. Math. Combin. Comput., 45 (2003), 3-19.

[16]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly even code invariant under M$^c$L:2, J. Combin. Theory Ser. A, 110 (2005), 53-69. doi: 10.1016/j.jcta.2004.10.001.

[17]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly-even code invariant under M$^c$L, Ars Combin., 91 (2009), 321-332.

[18]

C. Parker, E. Spence and V. D. Tonchev, Designs with the symmetric difference property on 64 points and their groups, J. Combin. Theory Ser. A, 67 (1994), 23-43. doi: 10.1016/0097-3165(94)90002-7.

[19]

R. Peeters, Uniqueness of strongly regular graphs having minimal $p$-rank, Linear Algebra Appl., 226/228 (1995), 9-31. doi: 10.1016/0024-3795(95)00184-S.

[20]

B. G. Rodrigues, "Codes of Designs and Graphs from Finite Simple Groups,'' Ph.D. thesis, University of Natal, Pietermaritzburg, 2002.

[21]

D. E. Taylor, "The Geometry of the Classical Groups,'' Heldermann Verlag, Berlin, 1992.

[22]

V. D. Tonchev, Hadamard matrices of order $36$ with automorphisms of order $17$, Nagoya Math. J., 104 (1986), 163-174.

[23]

R. A. Wilson, R. A. Parker and J. N. Bray, "Atlas of Finite Group Representations,'' available online at http://brauer.maths.qmul.ac.uk/Atlas/alt/A8/

show all references

References:
[1]

E. F. Assmus, Jr. and J. D. Key, "Designs and Their Codes,'' Cambridge University Press, 1992.

[2]

E. F. Assmus, Jr. and J. D. Key, Hadamard matrices and their designs: a coding-theoretic approach, Trans. Amer. Math. Soc., 330 (1992), 269-293. doi: 10.2307/2154164.

[3]

W. Bosma and J. Cannon, Handbook of Magma Functions, Department of Mathematics, University of Sydney, November 1994, available online at http://magma.maths.usyd.edu.au/magma

[4]

R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122. doi: 10.1112/blms/18.2.97.

[5]

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, "Atlas of Finite Groups,'' Oxford University Press, Oxford, 1985.

[6]

D. Crnković, V. Mikulić and S. Rukavina, Block designs constructed from the group U$(3,3)$, J. Appl. Algebra Discrete Struct., 2 (2004), 69-81.

[7]

U. Dempwolff, Primitive rank-$3$ groups on symmetric designs, Des. Codes Crypt., 22 (2001), 191-207. doi: 10.1023/A:1008373207617.

[8]

L. E. Dickson, "Linear Groups with an Exposition of the Galois Field Theory,'' Dover Publications, New York, 1958.

[9]

M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, available online at http://www.codetables.de

[10]

W. H. Haemers, R. Peeters and J. M. van Rijckevorsel, Binary codes of strongly regular graphs, Des. Codes Crypt., 17 (1999), 187-209. doi: 10.1023/A:1026479210284.

[11]

C. Jansen, K. Lux, R. Parker and R. Wilson, "An Atlas of Brauer Characters,'' with Appendix 2 by T. Breuer and S. Norton, The Clarendon Press, Oxford University Press, New York, 1995.

[12]

W. M. Kantor, Symplectic groups, symmetric designs, and line ovals, J. Algebra, 33 (1975), 43-58. doi: 10.1016/0021-8693(75)90130-1.

[13]

J. D. Key and J. Moori, Codes, designs and graphs from the Janko groups $J_1$ and $J_2$, J. Combin. Math. Combin. Comput., 40 (2002), 143-159.

[14]

J. D. Key and J. Moori, Correction to: Codes, designs and graphs from the Janko groups $J_1$ and $J_2$, [J. Combin. Math. Combin. Comput., 40 (2002), 143-159], J. Combin. Math. Combin. Comput., 64 (2008), 153.

[15]

J. D. Key, J. Moori and B. G. Rodrigues, On some designs and codes from primitive representations of some finite simple groups, J. Combin. Math. Combin. Comput., 45 (2003), 3-19.

[16]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly even code invariant under M$^c$L:2, J. Combin. Theory Ser. A, 110 (2005), 53-69. doi: 10.1016/j.jcta.2004.10.001.

[17]

J. Moori and B. G. Rodrigues, A self-orthogonal doubly-even code invariant under M$^c$L, Ars Combin., 91 (2009), 321-332.

[18]

C. Parker, E. Spence and V. D. Tonchev, Designs with the symmetric difference property on 64 points and their groups, J. Combin. Theory Ser. A, 67 (1994), 23-43. doi: 10.1016/0097-3165(94)90002-7.

[19]

R. Peeters, Uniqueness of strongly regular graphs having minimal $p$-rank, Linear Algebra Appl., 226/228 (1995), 9-31. doi: 10.1016/0024-3795(95)00184-S.

[20]

B. G. Rodrigues, "Codes of Designs and Graphs from Finite Simple Groups,'' Ph.D. thesis, University of Natal, Pietermaritzburg, 2002.

[21]

D. E. Taylor, "The Geometry of the Classical Groups,'' Heldermann Verlag, Berlin, 1992.

[22]

V. D. Tonchev, Hadamard matrices of order $36$ with automorphisms of order $17$, Nagoya Math. J., 104 (1986), 163-174.

[23]

R. A. Wilson, R. A. Parker and J. N. Bray, "Atlas of Finite Group Representations,'' available online at http://brauer.maths.qmul.ac.uk/Atlas/alt/A8/

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