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Many generator matrices for constructing extremal binary self-dual codes ofdifferent lengths have the form G=(I|A), where I is the n by n identity matrixand A is the n by n matrix fully determined by the first row. In this work, wedefine a generator matrix in which A is a block matrix, where the blocks comefrom group rings and also, A is not fully determined by the elements appearingin the first row. By applying our construction over F_2+uF_2 and by employingthe extension method for codes, we were able to construct new extremal binaryself-dual codes of length 68. Additionally, by employing a generalisedneighbour method to the codes obtained, we were able to construct many newbinary self-dual [68,34,12]-codes with the rare parameters gamma=7,8 and 9 inW_{68,2}. In particular, we find 92 new binary self-dual [68,34,12]-codes.

New Self-Dual Codes of length 68 from a 2 by 2 block matrix Construction and Group Rings