Tetracode construction

Definition 6.6.3   The tetracode is the $GF(3)$ code $T$ with elements

$$(0,0,0,0),\ (1,0,1,2),\ (1,2,0,1),\ (1,1,2,0),$$

$$(0,1,1,1),\ (2,0,2,1),\ (2,1,0,2),\ (2,2,1,0),\ (0,2,2,2).$$

It is a self-dual $(4,2,3)$ code.

Here is Conway's ``tetracode'' construction of the $C_{12}$. Represent each $12$-tuple $c=(c_1,...,c_{12}) \in C_{12}$ as a $3\times 4$ array

$$c= \left( \begin{array}{cccc} c_1 & c_2 & c_3 & c_4\\ c_5 &... ..._7 & c_8\\ c_9 & c_{10} & c_{11} & c_{12} \end{array}\right).$$

The projection of $c$ is

$$pr(c)=(c_5-c_9,c_6-c_{10},c_7-c_{11},c_8-c_{12}).$$

The row score is the sum of the elements in that row. The column score is the sum of the elements in that column. Of course, all computations are in $GF(3)$.

Lemma 6.6.4 (Conway)   An array $c$ is in $C_{12}$ if and only if

• the (common) score of all four columns equals the negative of the score of the top row.

• $pr(c)\in T$.

There are several facts one can derive from this construction.

There are $264$ codewords of weight $6$, $440$ codewords of weight $9$, $24$ codewords of weight $12$, and codewords $729$ total.

Pick an arbitrary subset of $9$ elements taken from $\{1,2,3,...,12\}$. It is the support of exactly two codewords of weight $9$ in $C_{12}$. Pick a random subset $S$ of $6$ elements taken from $\{1,2,3,...,12\}$. The probablity that $S$ is the support of some codeword of weight $6$ in $C_{12}$ is $1/7$.

Lemma 6.6.5   For each weight $6$ codeword $c$ in $C_{12}$, there is a weight $12$ codeword $c'$ such that $c+c'$ has weight $6$.

If we call $c+c'$ a ``complement'' of $c$ then ``the complement'' is unique up to sign.

proof: The support of the codewords of weight $6$ form a $S(5,6,12)$ Steiner system. Therefore, to any weight $6$ codeword $c$ there is a codeword $c''$ whose support is in the complement of that of $c$. let $c'=c''-c$. $\Box$

Remark 6.6.6   Although we shall not need it, it appears that for each weight $9$ codeword $c$ in $C_{12}$, there is a weight $12$ codeword $c'$ such that $c+c'$ has weight $6$.