New extremal binary self-dual codes of length 68 from generalized neighbors (Q1994939)

The team of researchers that wrote this work focus their attention on the concept of neighbour of a self-dual code (SD). A binary code is SD if equal to its dual code. Two SD codes \(C_1\) and \(C_2\) of equal length \(n\) are called neighbors if their intersection \(C_1\cap C_2\) is a code of almost the same dimension, i.e. \(\dim(C_1\cap C_2)=\frac{n}{2}-1\). The authors generalize the concept of neighbours with ``\(k\)-neighbors'' that are the following: Two SD codes \(C_1\), \(C_2\) are said to be \(k\)-range neighbors if \(d_N (C_1, C_2)=\frac{n}{2}-\dim(C_1\cap C_2)\leq k\) and they are called \(k\)-neighbors if \(d_N(C_1, C_2) = k\). After proving some theoretical results about the distance function \(d_N\), the research is further applied to construct extremal SD codes of length 68 from a given SD code. In particular, a total of 143 new extremal binary SD codes of length 68 with new weight enumerators, including the first known examples with parameters \(\gamma = 8, 9\) in the weight enumerator \(W_{68,2}\).