Numerical Results and Asymptotic Lower Bound on the Covering Radius of Reed-Muller Codes RM (2, 11) and RM (3, n)

抄録

The covering radius of the r-th order Reed-Muller code RM (r, n), denoted by ρ (r, n), is the maximum r-th order nonlinearity of n-variable Boolean functions. Using the Fourquet-Tavernier list-decoding algorithm and the Fourquet list-decoding algorithm, we discover, among monomial Boolean functions, 11-variable Boolean functions with second-order nonlinearity 856, and we determine that the covering radius of RM (3, 8) in RM (4, 8) is 56. Besides, it is proved that the complexity of the Fourquet algorithm for list decoding RM (r, n) is linear in the length of the code 2ⁿ given the decoding radius up to the Johnson bound. In this paper, we prove that the complexity of the Fourquet algorithm is also linear in 2ⁿ in some special cases when the decoding radius is close to 2^n-r. Moreover, following from the Carlet's method, we improve the best proven lower bound on the third-order nonlinearity of monomial Boolean functions. In a word, the original idea of our work is to improve the lower bound on ρ (r, n) according to two categories as follows: for small r and n, we search an n-variable Boolean function with larger r-th order nonlinearity using a list-decoding algorithm for Reed-Muller codes; for large n, we study a class of quartic monomial Boolean functions to improve the best proven lower bound on its third-order nonlinearity.