Minimal codes form a special class of linear codes characterized by the property that none of the codewords is covered by some other linearly independent codeword. In 1998, Ashikhmin and Barg gave a sufficient condition for a code to be minimal. There were no known examples of infinite families of minimal codes violating the Ashikhmin-Barg condition until a recent breakthrough by Ding et al. in 2018.

 

In this talk, we present general methods for constructing infinite families of minimal binary codes based on natural concepts related to Boolean functions such as the direct sum (bent concatenation) and the use of derivatives. Moreover, we introduce non-covering permutations that allow us to construct minimal codes violating the Ashikhmin-Barg condition. If time permits, we’ll discuss generalizations of this framework to the non-binary case. This is joint work with E. Pasalic, Y. Wei, and F. Zhang.