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Ternary codes from some reflexive uniform subset graphs

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Abstract

We examine the ternary codes \(C_3(A_i+I)\) from matrices \(A_i+I\) where \(A_i\) is an adjacency matrix of a uniform subset graph \(\Gamma (n,3,i)\) of \(3\)-subsets of a set of size \(n\) with adjacency defined by subsets meeting in \(i\) elements of \(\Omega \), where \(0 \le i \le 2\). Most of the main parameters are obtained; the hulls, the duals, and other subcodes of the \(C_3(A_i+I)\) are also examined.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of the Western Cape, 7535 , Bellville, South Africa

    W. Fish, J. D. Key & E. Mwambene

Authors
  1. W. Fish
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  2. J. D. Key
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  3. E. Mwambene
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Correspondence to J. D. Key.

Appendix

Appendix

The general description of the words \(w_\pi \) that sum to give \(w_1-w_2\) when \(n \equiv 0 \text{(mod } 3\text{) }\) used in the proof of Proposition 6 is given here. For short, we write \([i,j,k,l]\) to denote the word \(w_\pi \) from the partition \([[1,2],[i,j],[k,l]]\) as defined in Definition 2. We take the sum of:

$$\begin{aligned}&[3,4,5,6],[3,5,4,6];[3,7,8,9],[3,8,7,9];\ldots ;\\&[3,n-2,n-1,n],[3,n-1,n-2,n];\\&[4,7,8,9],[4,8,7,9];\ldots ;[4,n-2,n-1,n],[4,n-1,n-2,n];\\&[5,7,8,9],[5,8,7,9];\ldots ;[5,n-2,n-1,n],[5,n-1,n-2,n];\\&[6,7,8,9],[6,8,7,9];\ldots ;[6,n-2,n-1,n],[6,n-1,n-2,n];\\&[7,10,11,12],[7,11,10,12];\ldots ;[7,n-2,n-1,n],[7,n-1,n-2,n];\\&[8,10,11,12],[8,11,10,12];\ldots ;[8,n-2,n-1,n],[8,n-1,n-2,n];\\&[9,10,11,12],[9,11,10,12];\ldots ;[9,n-2,n-1,n],[7,n-1,n-2,n];\\&\vdots \\&[n-5,n-2,n-1,n],[n-5,n-1,n-2,n];\\&[n-4,n-2,n-1,n],[n-4,n-1,n-2,n];\\&[n-3,n-2,n-1,n],[n-3,n-1,n-2,n]; \end{aligned}$$

It is easy to verify that these \(\frac{1}{3}(n-3)(n-4)\) words \(w_\pi \) add up to \(w_1-w_2\).

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Fish, W., Key, J.D. & Mwambene, E. Ternary codes from some reflexive uniform subset graphs. AAECC 25, 363–382 (2014). https://doi.org/10.1007/s00200-014-0233-4

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  • DOI: https://doi.org/10.1007/s00200-014-0233-4

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