From Robert Griess's article "Elementary abelian p-subgroups of algebraic groups":
(2.7) Definition. Let char$(\mathbb{K})\neq 2$ and let $V$ be an $m$-dimensional vector space with nondegenerate binlinear form $f$ and orthonormal basis $B$= { $e_{i}|i \in \Omega$ } , where $\Omega$= {$1,...,m$} . We call $B$ a frame. A signed frame is a set of the form $\pm B $. The associated frame group is the subgroup of Aut($f$) stabilizing $\pm B $; it is isomorphic to $2$ wr $S_m$. The frame group is a monomial group which is a semidirect product of the diagonal frame group $D$ and the group of permutation matrices. The elements $v_i$ of $D$ which satisfy $e_{i}v_j=e_i$ or $-e_i$ as $j\neq i$ and $j=i$ form a basis for $D$ and give $D$ the structure of $F_{2}^{n}$; a subgroup of $D$ is naturally identified with a code, so the code-theoretic notions of weight, etc,. apply to $D$.
Let's say $V$ is 8-dimensional. What are the $nearly$ self-orthogonal even code in $D\cong 2^8 $.
Table V of this paper gives the classification of small-dimensional nearly self-orthogonal binary codes and their groups:
| Length | Dimension | Basis | Group |
|---|---|---|---|
| 2 | 1 | 12 | $S_2$ |
| 3 | 1 | 12 | $S_3$ |
| 4 | 2 | 12,34 | 2 wr 2 |
| 5 | 2 | 12,34 | 2 wr 2 |
| 6 | 3 | 12,34,56 | 2 wr $S_3$ |
May I get an example of how to pick from the table to get me going? I have a tiny bit knowledge of basic definitions of coding theory.
Edit: I guess this boils down to how to figure out if the codes listed in the table are even or not. Where should I get the info about their parity-check matrices? Thx!