Bent Vectorial Functions, Codes and Designs

Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group (\gf(2^{2m}), +), have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus-Mattson theorem, but nevertheless hold $2$-designs. A new coding-theoretic characterization of bent vectorial functions is presented

The Extended Codes of Some Linear Codes

The classical way of extending an $[n, k, d]$ linear code \C is to add an overall parity-check coordinate to each codeword of the linear code \C. This extended code, denoted by \overline{\C}(-\bone) and called the standardly extended code of \C, is a linear code with parameters $[n+1, k, \bar{d}]$, where $\bar{d}=d$ or $\bar{d}=d+1$. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper