The Existence Of The Optimal Impulse Radio Sequences With Weight 5

Ultra Wide Band(UWB)is known as Impulse Radio,which is a new kind of wireless communication equipments with low cost,low consumption,high transmission rate,etc.So it can be applied to wireless multimedia communication,precision positioning,radar imaging and other fields.The concept of impulse radio sequences was firstly introduced by Chu and Colbourn in 2004 to study sequences or signal designs for ultra wide band radio with unmodulated time-hopping.The impulse radio sequence is closely related to the optical orthogonal code(OOC)and strict cyclic packing design(SCP),which is concerned by many scholars in the field of inf ormation theory and combinational design.Let m,?,?a,?c be positive integers.An(m,?,?a,?c)impulse radio sequence,briefly denoted by(m,?,?a,?c)-IRS,is a family C of(0,1)sequences of length ?m and weight? satisfying the pulse position property,the autocorrelation coefficient ?a and the cross-correlation coefficient ?c.Let(m,?,?a,?c)be the largest possible number of codewords among all(m,?,?a,?c)-IRSs.An(m,?,?a,?c)-IRS with ?(m,?,?a,?c)codewords is said to be optimal(or maximum).The central of researches on IRSs lies in determin-ing the value)(m,?,?a,?c)and constructions of the optimal IRSs.The exact values of ?(m,?,?—1)and?(m,?,?a,?c—1)have been completely determined.Subsequently,the upper bounds of ?(m,?,?,1)and ?(m,?,?)were given by Chang,etc.,when 2????—1.For ?=3,4,the exact values of ?(m,?,?a,?—1)have already been got;For ?= 5,the exact value of?(m,5,3,4)has also been given out.In this paper,we will use two different methods to discuss the optimal values of the impulse radio sequences with ?= 5,?c= 4,?a=2,1.For ?(m,5,2,4),we firstly give the forms and classifications of all the orbits,then we use the knowledge of combinational design and number theory to calculate?(m,5,2,4).According to the definition of the impulse radio sequences,the problem of ?(m,5,1,4)is transformed into the number of the 4 tuples with corresponding conditions.Then we can get the exact value of ?(m,5,1,4)by using the inclusion-exclusion principle repeatedly.