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On the non-minimality of the largest weight codewords in the binary Reed-Muller codes
May 2011, 5(2): 339-350. doi: 10.3934/amc.2011.5.339
Some optimal codes related to graphs invariant under the alternating group $A_8$
Bernardo Gabriel Rodrigues 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.
Keywords: automorphism group., optimal linear code, Symmetric designs, self-orthogonal codes.
Mathematics Subject Classification: Primary: 05C45, 05B05, 20D45; Secondary: 94B0.
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
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Bernardo Gabriel Rodrigues