turkmath.org

Extension theorems are generalized to self-dual codes over rings and as applications many new binary self-dual extremal codes are found from self-dual codes over $\mathbb{F}_{2^{m}}+u\mathbb{F}_{2^{m}}$ for m=1,2. The duality and distance preserving Gray maps from $\mathbb{F}_{4}+u\mathbb{F}_{4}$ to $\left( \mathbb{F}_{2}+u\mathbb{F}_{2}\right) ^{2}$ and $\mathbb{F}_{2}^{4}$ are used to obtain self-dual codes whose binary Gray images are [64,32,12]-extremal self-dual. The $\mathbb{F}_{2}+u\mathbb{F}_{2}$-codes are extended over $\mathbb{F}_{2}+u\mathbb{F}_{2}$ and as binary Gray images of the extensions 178 extremal binary self-dual codes of length 68 are obtained with new weight enumerators. Especially the first examples of codes with $\gamma =3$ and many codes with the rare $\gamma =4,6$ parameters are obtained. New extremal doubly even binary codes of lengths 80 and 88 are also found by the $\mathbb{F}_{2}+u\mathbb{F}_{2}$-lifts of binary four circulant codes with a corresponding result about 3-designs.