Construction of covering arrays from Outline m-sequences Covering - PowerPoint PPT Presentation

Construction of covering arrays from m-sequences Georgios Tzanakis Construction of covering arrays from Outline m-sequences Covering arrays Definition Research on CAs Motivation Sequences Georgios Tzanakis 1 Definition m-sequences Our work Joint work with L. Moura 2 and D. Panario 1 In a nutshell Our method Current results Future Carleton University 1 University of Ottawa 2 December 5, 2013 1 / 38

Construction of covering arrays from m-sequences Georgios Tzanakis Outline Covering arrays Definition Research on CAs Motivation Sequences Definition m-sequences Our work In a nutshell Our method Current results Future WORK IN PROGRESS 2 / 38

Construction of Outline of talk covering arrays from m-sequences Georgios Tzanakis Covering arrays Outline Definition Covering arrays Research on covering arrays Definition Research on CAs Motivation Motivation Sequences Definition Linear recurrence sequences over finite fields m-sequences Our work Definition In a nutshell Our method m-sequences Current results Future Our work In a nutshell Our method Current results Future 3 / 38

Construction of Outline of talk covering arrays from m-sequences Georgios Tzanakis Covering arrays Outline Definition Covering arrays Research on covering arrays Definition Research on CAs Motivation Motivation Sequences Definition Linear recurrence sequences over finite fields m-sequences Our work Definition In a nutshell Our method m-sequences Current results Future Our work In a nutshell Our method Current results Future 4 / 38

Construction of Definition of covering arrays covering arrays from m-sequences Georgios Tzanakis A covering array CA ( N ; t , k , v ) is a N × k array with entries Outline Covering arrays from an alphabet of size v , with the property that any N × t Definition sub-array has at least one row equal to every possible t -tuple. Research on CAs Motivation Sequences Definition m-sequences Our work In a nutshell Our method Current results Future 5 / 38

Construction of Definition of covering arrays covering arrays from m-sequences Georgios Tzanakis A covering array CA ( N ; t , k , v ) is a N × k array with entries Outline Covering arrays from an alphabet of size v , with the property that any N × t Definition sub-array has at least one row equal to every possible t -tuple. Research on CAs Motivation Sequences Definition m-sequences 0 0 0 0 0 0 0 0 0 0 Our work 1 1 1 1 1 1 1 1 1 1 In a nutshell 1 1 1 0 1 0 0 0 0 1 Our method 1 0 1 1 0 1 0 1 0 0 Current results Example Future 1 0 0 0 1 1 1 0 0 0 A covering array 0 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 1 0 CA (13; 3 , 10 , 2) 1 1 0 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 5 / 38

Construction of Definition of covering arrays covering arrays from m-sequences Georgios Tzanakis A covering array CA ( N ; t , k , v ) is a N × k array with entries Outline Covering arrays from an alphabet of size v , with the property that any N × t Definition sub-array has at least one row equal to every possible t -tuple. Research on CAs Motivation Sequences Definition m-sequences 0 0 0 0 0 0 0 0 0 0 Our work 1 1 1 1 1 1 1 1 1 1 In a nutshell 1 1 1 0 1 0 0 0 0 1 Our method 1 0 1 1 0 1 0 1 0 0 Current results Example Future 1 0 0 0 1 1 1 0 0 0 A covering array 0 1 1 0 0 1 0 0 1 0 0 0 1 0 1 0 1 1 1 0 CA (13; 3 , 10 , 2) 1 1 0 1 0 0 1 0 1 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 1 5 / 38

Construction of Definition of covering arrays covering arrays from m-sequences Georgios Tzanakis A covering array CA ( N ; t , k , v ) is a N × k array with entries Outline Covering arrays from an alphabet of size v , with the property that any N × t Definition sub-array has at least one row equal to every possible t -tuple. Research on CAs Motivation Sequences Definition m-sequences 0 0 0 0 Our work In a nutshell 0 1 2 2 Our method Current results 1 2 2 0 Future Example 2 2 0 2 A covering array 2 0 2 1 CA (9; 2 , 4 , 3) 0 2 1 1 2 1 1 0 1 1 0 1 1 0 1 2 5 / 38

Construction of Definition of covering arrays covering arrays from m-sequences Georgios Tzanakis A covering array CA ( N ; t , k , v ) is a N × k array with entries Outline Covering arrays from an alphabet of size v , with the property that any N × t Definition sub-array has at least one row equal to every possible t -tuple. Research on CAs Motivation Sequences Definition m-sequences 0 0 0 0 Our work In a nutshell 0 1 2 2 Our method Current results 1 2 2 0 Future Example 2 2 0 2 A covering array 2 0 2 1 CA (9; 2 , 4 , 3) 0 2 1 1 2 1 1 0 1 1 0 1 1 0 1 2 5 / 38

Construction of Outline of talk covering arrays from m-sequences Georgios Tzanakis Covering arrays Outline Definition Covering arrays Research on covering arrays Definition Research on CAs Motivation Motivation Sequences Definition Linear recurrence sequences over finite fields m-sequences Our work Definition In a nutshell Our method m-sequences Current results Future Our work In a nutshell Our method Current results Future 6 / 38

Construction of Research on covering arrays covering arrays from m-sequences Georgios Tzanakis Outline Covering arrays Definition Research on CAs Motivation Sequences 1. Bounds on number of rows Definition m-sequences 2. Combinatorial and algebraic constructions Our work In a nutshell 3. Computer-generated constructions Our method Current results 4. Recursive constructions Future 7 / 38

Construction of Research on covering arrays covering arrays from m-sequences Georgios Tzanakis Outline Covering arrays Definition Research on CAs Motivation Sequences 1. Bounds on number of rows Definition m-sequences 2. Combinatorial and algebraic constructions Our work In a nutshell 3. Computer-generated constructions Our method Current results 4. Recursive constructions Future 7 / 38

Construction of Research on covering arrays covering arrays from m-sequences 1. Bounds on number of rows Georgios Tzanakis Outline Covering arrays Definition Definition Research on CAs Motivation The covering array number CAN ( t , k , v ) is the smallest Sequences possible N such that a CA ( N ; t , k , v ) exists Definition m-sequences Our work Colbourn, ’04 In a nutshell Our method “Lower bounds are in general not well explored. . . ” Current results Future 8 / 38

Construction of Research on covering arrays covering arrays from m-sequences 1. Bounds on number of rows Georgios Tzanakis Outline Covering arrays Definition Definition Research on CAs Motivation The covering array number CAN ( t , k , v ) is the smallest Sequences possible N such that a CA ( N ; t , k , v ) exists Definition m-sequences Our work Elementary counting arguments In a nutshell Our method ◮ v t ≤ CAN ( t , k , v ) ≤ v k Current results Future ◮ CAN ( t − 1 , k − 1 , v ) ≤ 1 v CAN ( t , k , v ) ◮ If k 1 < k 2 then CAN ( t , k 1 , v ) < CAN ( t , k 2 , v ) ◮ . . . 8 / 38

Construction of Research on covering arrays covering arrays from m-sequences 1. Bounds on number of rows Georgios Tzanakis Outline Covering arrays Definition Definition Research on CAs Motivation The covering array number CAN ( t , k , v ) is the smallest Sequences possible N such that a CA ( N ; t , k , v ) exists Definition m-sequences Our work Case t = 2 , v = 2 In a nutshell Our method Current results ◮ Kleitman and Spencer ’73; Katona ’73 Future � � N − 1 �� CAN (2 , k , 2) = min N ∈ N ; k ≤ ⌈ N 2 ⌉ 8 / 38

Construction of Research on covering arrays covering arrays from m-sequences 1. Bounds on number of rows Georgios Tzanakis Outline Covering arrays Definition Definition Research on CAs Motivation The covering array number CAN ( t , k , v ) is the smallest Sequences possible N such that a CA ( N ; t , k , v ) exists Definition m-sequences Our work Case t = 2 , v > 2 In a nutshell Our method Current results ◮ Gargano, K¨ orner, Vacarro ’90 Future CAN (2 , k , v ) = v 2 log K (1 + o (1)) 8 / 38

Construction of Research on covering arrays covering arrays from m-sequences 1. Bounds on number of rows Georgios Tzanakis Outline Covering arrays Definition Definition Research on CAs Motivation The covering array number CAN ( t , k , v ) is the smallest Sequences possible N such that a CA ( N ; t , k , v ) exists Definition m-sequences Our work Recursive results In a nutshell Our method ◮ CAN (2 , kq + 1 , q ) ≤ CAN (2 , k , q ) + q 2 − q Current results Future ◮ CAN (2 , k ( q + 1) , q ) ≤ CAN (2 , k , q ) + q 2 − 1 ◮ CAN (3 , 2 k , v ) ≤ CAN (3 , k , v ) + ( v − 1) CAN (2 , k , v ) ◮ . . . 8 / 38