Papers for
error correction system designers
Papers whose findings have a practical use for this group, as judged from the abstract. Open a paper to read what it means in practice.
Maximum nonlinearity of balanced 8-variable Boolean functions fixed at 116
The Maximum Nonlinearity of 8-variable Balanced Boolean Functions is 116
Abstract: Does there exist an 8-variable balanced Boolean function with nonlinearity 118? This is a famous and long-standing open problem in Boolean functions. In this paper, we give a negative answer to this open problem, and prove that the maximum nonlinearity of balanced Boolean functions in 8 variables is 116, establishing the 8-variable instance of Dobbertin's conjectured recursion for the minimum Walsh peak of balanced functions. Equivalently, the covering radius of the Reed--Muller code RM(1,8) in 0-resilient Boolean functions is 116. We deduce the result by obtaining a quadratic form of polar rank 6, and reducing the problem to the existence of a 6-variable Boolean function with nonlinearity greater than 26 whose Hamming distance to a bent function is between 4 and 6.
New limits found for unique decoding in code pairs communication
A Continuous Projection Converse and a Weighted Construction for Uniquely Decodable Code Pairs
Abstract: We study the maximum sum rate of uniquely decodable code pairs. A weighted refinement of a complement-gluing construction yields an explicit code of length 672 and sum rate exceeding $1.318639029203$. An iterated projection argument, combined with a conditional entropy estimate and a justified continuous limit, gives an upper bound of $1.480063425539$.