Question

The Parker loop has order $2^{13}$, and reducing it modulo its 'center' yields -- not just a $12$-dimensional $\mathbb{Z} /(2)$-vector space, but one which can be naturally identified with the subspace of $(\mathbb{Z} /(2))^{24}$ which is the extended $24$-bit Golay code.

Analogous to the extended $24$-bit Golay code is the $12$-coordinate extended ternary Golay code, which is a $6$-dimensional subspace of $(\mathbb{Z} /(3))^{12}$. Is there likewise a known analogue of the Parker loop which has order $3^{7}$ and which may be likewise identified with the extended ternary Golay code once it quotiented by its center? (And, if no such thing is known, is it because it has been proven not to exist?)

Answer 1

Parker's loop can be viewed as a special case of Griess's construction of code loops for doubly even binary codes ("Code loops", J. Algebra 100 (1986), 224-234, http://deepblue.lib.umich.edu/bitstream/2027.42/26195/1/0000274.pdf). Over $F_p$ with $p$ odd, things are more subtle, but there are some constructions.

For example, see "Local subgroups of the Monster and odd code loops" by Thomas Richardson (Trans. AMS 247 (1995), 1453-1531, http://www.ams.org/journals/tran/1995-347-05/S0002-9947-1995-1266532-4/S0002-9947-1995-1266532-4.pdf). His construction gives a loop of order $3^7$ corresponding to the ternary Golay code.

The analogy may not be perfect: for example, the loop Richardson constructs has a slightly bigger center than one might guess. However, these odd code loops seem to be useful for some of the same sort of things as Parker's loop is. I don't know whether there is an even closer analogue (it's not clear to me which properties are really essential).

Answer 2

To follow up on Henry Cohn's answer, see also:

A. Drapal and P. Vojtechovsky, Code loops in both parities, Journal of Algebraic Combinatorics 31 (2010), no. 4, 585-611.