Extension theorems for self-dual codes over rings and new binary self-dual codes

In this work, 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 F2m + uF2m for m = 1, 2. The duality and distance preserving Gray maps from F4 + uF4 to (F2 + uF2) and F42 are used to obtain self-dual codes whose binary Gray images are [64, 32, 12]-extremal self-dual. An F2 + uF2-extension is used and as binary images, 178 extremal binary self-dual codes of length 68 with new weight enumerators are obtained. Especially the first examples of codes with γ = 3 and many codes with the rare γ = 4, 6 parameters are obtained. In addition to these, two hundred fifty doubly even self dual [96, 48, 16]-codes with new weight enumerators are obtained from four-circulant codes over F4 + uF4. New extremal doubly even binary codes of lengths 80 and 88 are also found by the F2+uF2-lifts of binary four circulant codes and a corresponding result about 3-designs is stated.