Ternary codes from some reflexive uniform subset graphs

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.

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 $w_1-w_2$.