Self-dual codes which are principal ideals of the group algebra \(\mathbb{F}_ 2[\{\mathbb{F}_{2^ m},+\}]\) (Q1361703)

The author studies Camion's \(H\)-codes, self dual codes that are principal ideals in the group algebra \(\mathbb{F}_2 [\{\mathbb{F}_{2^m}, +\}]\). In particular, an upper bound is derived on the minimum distance of \(H\)-codes. As a corollary, if \(d_m\) denotes the minimum distance of any binary \(H\)-code of length \(2^m\), then \(\lim_{m \to\infty}d_m/2^m =0\). Moreover, it is shown that no \(H\)-codes of length \(>32\) meet Gleason's bound, and it is proved that extremal [32,16] \(H\) codes are characterized by quadratic bent functions over \(\mathbb{F}_{16}\), corresponding to the words of weight 6 or 10 of the \((2,4)RM\) code.