## The theory of 2-functions

dc.contributor.author | Johns, Bryan R. | |

dc.date.accessioned | 2016-04-05T19:11:54Z | |

dc.date.available | 2016-04-05T19:11:54Z | |

dc.date.issued | 2013 | |

dc.identifier.isbn | 9781303751004 | |

dc.identifier.other | 1509130670 | |

dc.identifier.uri | http://hdl.handle.net/10477/50361 | |

dc.description.abstract | This paper contains a description of the cubic homogeneous 2-monomial rotation symmetric (2-MRS) Boolean functions, that is, Boolean functions of degree 3 which are generated by applying even powers of the cyclic rotation to a single monomial. We begin by dividing such functions into two classes: pure form functions, which contain only odd-indexed variables, and mixed form functions, which contain variables of both even and odd index. We prove that the pure form functions are equivalent to the standard monomial rotation symmetric Boolean functions, which have been described in previous work. We then turn our attention to the mixed form functions and remain focused on these for the remainder of the paper. We define the χ -value of a given mixed form function to be the difference between the two indexes in the function's generating monomial which are equivalent modulo 2. With this definition in mind, we prove that functions which have the same χ -value are affine equivalent. Next, we extend the work of Brown and Cusick (which itself is an extension of the work of Bileschi, Cusick, and Padgett) to derive recursions for the weights of the cubic 2-MRS functions. Based on the equivalences given by the χ -values of these functions, we refine the construction of the "rules" matrices for these recursions and find that they all have a predictable form. We then discuss the recursions themselves, whose recursion polynomials (characteristic polynomials) have special forms. We analyze these polynomials in depth and in so doing prove that the equivalence classes that we found before (those that are produced by permutations which preserve 2-rotation symmetry) are, in fact, the equivalence classes under arbitrary transformations. We then define the idea of "crucial" weights/Walsh numbers, which are the weights/Walsh numbers of a 2-function in the number of variables equal to and integer multiple of the function's χ -value. We derive a recursion for these crucial values and use it to derive a recursion for the weights of the 2-functions, generally (which verifies our previous results). We end by exploring a connection between the recursions for the 2-functions and the Lucas numbers. | |

dc.language | English | |

dc.source | Dissertations & Theses @ SUNY Buffalo,ProQuest Dissertations & Theses Global | |

dc.subject | Pure sciences | |

dc.subject | 2-functions | |

dc.subject | Boolean functions | |

dc.subject | Monomial rotation symmetric functions | |

dc.subject | Rotation symmetry | |

dc.title | The theory of 2-functions | |

dc.type | Dissertation/Thesis |