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The MINIMOG description

Following Curtis' description [Cu2] of a Steiner system $S(5,8,24)$ using a $4\times 6$ array, called the MOG, Conway [Co1] found and analogous description of $S(5,6,12)$ using a $3\times 4$ array, called the MINIMOG. This section is devoted to the MINIMOG. The tetracode words are
0 0 0 0 0 + + + 0 - - -
+ 0 + - + + - 0 + - 0 +
- 0 - + - + 0 - - - + 0
With $\lq\lq 0''=0,\ \lq\lq +''=1,\ \lq\lq -''=2$, these vectors form a linear code over $\mathbf{F}_3$. (This notation is Conway's. One must remember here that ``+''+``+''=``-'' and ``-''+``-''=``+''!) They may also be described as the set of all 4-tuples in $\mathbf{F}_3$ of the form

\begin{displaymath} (0,a,a,a),\ \ (1,a,b,c),\ \ (2,c,b,a), \end{displaymath}

where $abc$ is any cyclic permutation of $012$. The MINIMOG in the shuffle numbering is the $3\times 4$ array

\begin{displaymath} \begin{array}{cccc} 6 & 3 & 0 & 9\\ 5 & 2 & 7 & 10 \\ 4 & 1 & 8 & 11 \end{array} \end{displaymath}

We label the rows as follows:
0 6 3 0 9
+ 5 2 7 10
- 4 1 8 11
A col (or column) is a placement of three + signs in a column of the array:
+      
+      
+      
  +    
  +    
  +    
    +  
    +  
    +  
      +
      +
      +
A tet (or tetrad) is a placement of 4 + signs having entries corresponding (as explained below) to a tetracode.
+ + + +
       
       
+      
  + + +
       
+      
       
  + + +
0 0 0 0
0 + + +
0 - - -
  +    
+   +  
      +
      +
+ +    
    +  
    +  
+     +
  +    
+ 0 + -
+ + - 0
+ - 0 +
  +    
      +
+   +  
    +  
  +    
+     +
      +
    +  
+ +    
- 0 - +
- + 0 -
- - + 0
Each line in $\mathbf{F}_3^2$ with finite slope occurs once in the $3\times 3$ part of some tet. The odd man out for a column is the label of the row corresponding to the non-zero digit in that column; if the column has no non-zero digit then the odd man out is a ``?''. Thus the tetracode words associated in this way to these patterns are the odd men out for the tets. The signed hexads are the combinations $6$-sets obtained from the MINIMOG from patterns of the form
col-col, col+tet, tet-tet, col+col-tet.

Lemma 3 (Conway, [CS1], chapter 11, page 321)   If we ignore signs, then from these signed hexads we get the 132 hexads of a Steiner system $S(5,6,12)$. These are all possible $6$-sets in the shuffle labeling for which the odd men out form a part (in the sense that an odd man out ``?'' is ignored, or regarded as a ``wild-card'') of a tetracode word and the column distribution is not $0,1,2,3$ in any order 2.

Example 4   Associated to the col-col pattern
+      
+      
+      
-
  +    
  +    
  +    
=
+ -    
+ -    
+ -    
is the tetracode $0\ 0\ ?\ ?$ and the signed hexad $\{-1,-2,-3,4,5,6\}$ and the hexad $\{1,2,3,4,5,6\}$. Associated to the col+tet pattern
  +    
  +    
  +    
+
+      
  + + +
       
=
+ +    
  - + +
  +    
is the tetracode $0\ +\ +\ +$ and the signed hexad $\{1,-2,3,6,7,10\}$ and the hexad $\{1,2,3,6,7,10\}$.

Furthermore, it is known [Co1] that the Steiner system $S(5,6,12)$ in the shuffle labeling has the following properties.
Next: The shuffle kitten Up: MINIMOGs and Mathematical blackjack Previous: Curtis' kitten   Contents
David Joyner
2000-05-29