Asymptotically good four Toeplitz codes and derived new formally self-dual LCD codes

Lots of attention has recently been focused on isodual (resp. formally self-dual (FSD)) codes and linear complementary dual (LCD) codes for their theoretical and practical importance. In this paper, we consider a general construction, namely the class of four Toeplitz (FT) codes and prove that they arbitrarily approach the asymptotic Gilbert–Varshamov bound on the relative distance with asymptotic rate \(\frac{1}{2}\) $\frac{1}{2}$ over arbitrary finite fields. Then we study a special family of isodual and further FSD FT codes whose asymptotic behavior is the same as that of FT codes. In terms of eigenvalues, we further provide a sufficient and necessary condition for a family of FT codes to be LCD. In addition, we introduce bordered FT codes, which can be effectively tested for constructing isodual codes with longer lengths. Finally, we obtain many improved isodual LCD and further FSD LCD codes compared to the latest results reported in the literature.