TY - JOUR

T1 - The graph of minimal distances of bent functions and its properties

AU - Kolomeec, Nikolay

PY - 2017/12/1

Y1 - 2017/12/1

N2 - A notion of the graph of minimal distances of bent functions is introduced. It is an undirected graph (V, E) where V is the set of all bent functions in 2k variables and (f, g) ∈ E if the Hamming distance between f and g is equal to 2 k. It is shown that the maximum degree of the graph is equal to 2 k(2 1+ 1) (2 2+ 1) ⋯ (2 k+ 1) and all its vertices of maximum degree are quadratic bent functions. It is obtained that the degree of a vertex from Maiorana—McFarland class is not less than 2 2 k + 1- 2 k. It is proven that the graph is connected for 2 k= 2 , 4 , 6 , disconnected for 2 k≥ 10 and its subgraph induced by all functions EA-equivalent to Maiorana—McFarland bent functions is connected.

AB - A notion of the graph of minimal distances of bent functions is introduced. It is an undirected graph (V, E) where V is the set of all bent functions in 2k variables and (f, g) ∈ E if the Hamming distance between f and g is equal to 2 k. It is shown that the maximum degree of the graph is equal to 2 k(2 1+ 1) (2 2+ 1) ⋯ (2 k+ 1) and all its vertices of maximum degree are quadratic bent functions. It is obtained that the degree of a vertex from Maiorana—McFarland class is not less than 2 2 k + 1- 2 k. It is proven that the graph is connected for 2 k= 2 , 4 , 6 , disconnected for 2 k≥ 10 and its subgraph induced by all functions EA-equivalent to Maiorana—McFarland bent functions is connected.

KW - Affinity

KW - Bent functions

KW - Boolean functions

KW - The minimal distance

UR - http://www.scopus.com/inward/record.url?scp=85002489610&partnerID=8YFLogxK

U2 - 10.1007/s10623-016-0306-4

DO - 10.1007/s10623-016-0306-4

M3 - Article

AN - SCOPUS:85002489610

VL - 85

SP - 395

EP - 410

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

SN - 0925-1022

IS - 3

ER -