Higgsy Codes

Dharwadker's four color theorem uses the Steiner system $S(5,8,24)$, otherwise known as the Witt design. To construct the Steiner system, one starts with the extended binary Golay code. This has $2^{12} = 4096$ length $24$ words, selected from all possible binary words by writing a list of length $24$ binary words so that each word differs from the previous ones in at least $8$ places. It turns out that the code words are partitioned

$4096 = 1 + 759 + 2576 + 759 + 1$

into words with no $1$-bits, words with $8$ $1$-bits, words with $12$ $1$-bits, $16$ $1$-bits and $24$ $1$-bits, respectively. The $729$ words with $8$ $1$-bits form the Steiner system, which is defined as a set of $24$ points, along with the $759$ special subsets of size $8$, called blocks, such that every $5$ points is contained in one of the blocks. The number $759$ is very interesting. It can be partitioned into

$759 = 1 + 280 + 448 + 30$

which counts, respectively, the number of subsets that intersect a given subset in $8$, $4$, $2$ or zero elements. That is, there are $30$ blocks with no intersection with a given block, and $729 = 27^2$ with a non empty intersection. We write $280 = 4 \times 70$, $448 = 16 \times 28$ and $30 = 30 \times 1$, where $70$, $28$ and $1$ are weights for the Reed-Muller code $RM(2,3)$. The $4$, $16$ and $30$ appear in the bottom row of the Pascal triangle below. The number 759 is the summit of this triangle, which counts the number of blocks that (i) contain a point set with $n$ elements and (ii) are disjoint from another set with $m$ elements. The union cardinality $m + n$ is fixed in each row. So there are $253$ occurrences of any given object, and a complementary $506$ blocks that do not contain it.

There are $42504$ ways to choose $5$ objects from $24$, and $42504 = 759 \times 56$. To obtain the $8$ bits, after choosing the first $5$, there are $969$ possibilities (namely $3$ out of $19$) and we see that $969 \times 759$ is indeed the binomial coefficient $B(24,8) = 735471$. Why the redundancy factor of $56$, the FTS dimension? This counts the number of ways of choosing $3$ objects from $8$, namely the $3$ that are omitted from the set of $5$.

The Mathieu group $M_{24}$ has a 729 dimensional permutation representation with suborbit lengths given by the $1$, $30$, $280$ and $448$. Similarly, the Mathieu group $M_{12}$ is associated to the extended ternary Golay code. The two binary and two ternary Golay codes, with lengths $11$, $12$, $23$ and $24$, are the only binary and ternary codes with suitably nice properties. There are $729 = 27^2$ words in the ternary code of length $11$, and the extended ternary code also forms a Steiner system, on $12$ objects. The $12$ objects are thought of as a projective line over the $11$ element field $F_{11}$ with the addition of $\infty$. The blocks are built using fractional linear transformations, which we know how to cover with Jordan algebra braids.