On Isodual Double Toeplitz Codes

Double Toeplitz (shortly DT) codes are introduced here as a generalization of double circulant codes. The authors show that such a code is isodual, hence formally self-dual (FSD). FSD codes form a far-reaching generalization of self-dual codes, the most important class of codes of rate one-half. Self-dual DT codes are characterized as double circulant or double negacirculant. Likewise, even binary DT codes are characterized as double circulant. Numerical examples obtained by exhaustive search show that the codes constructed have best-known minimum distance, up to one unit, amongst formally self-dual codes, and sometimes improve on the known values. For q = 2, the authors find four improvements on the best-known values of the minimum distance of FSD codes. Over \(\mathbb{F}_{4}\) $${\mathbb{F}}_4$$ an explicit construction of DT codes, based on quadratic residues in a prime field, performs equally well. The authors show that DT codes are asymptotically good over \(\mathbb{F}_{q}\) $${\mathbb{F}}_q$$ . Specifically, the authors construct DT codes arbitrarily close to the asymptotic Varshamov-Gilbert bound for codes of rate one half.