Families of Four-phase Quasi-orthogonal Code Arrays

A method of constructzon is presented for rectangular quaszorthogonal code arrays over the ring Z+ The proposed arrays are easy to generate: The four-phase lznear recurrzng sequences constructed by Bozta8 et. al. are utilized to generate the arrays by modulo 4 subtractcon. The perzodzc autoand cross-correlataon propertaes of these arrays are then dersved in a strasghtforward manner. The maximum off-peak correlation magnitude for these arrays is lower b y a factor of 4 when compured to the binary Gold code arrays constructed b y Kuo and Rigas. The arrays can be used for 'add-on' data transmission, pattern synchronization, and code division image multiplexing. INTRODUCTION Kuo and Rigas introduced binary quasi m-arrays and quasi Gold arrays in [l]. These arrays were proposed to overcome some disadvantages associated with the m-arrays studied by Nomura et. al. [2] and lliams and Sloane [3]. Their construction for quasi m-arrays yields L1 x L2 binary arrays where L, = 2"' 1, with n, positive integers for z = 1 , 2 . Given two binary sequences, say s l ( t ) , t = 0,1,. . . , L1 1 and s2(t) , t = 0,1,. . ., L2 1 (these sequences can either be two msequences or two Gold sequences) Kuo and Rigas use the construction a(ti,tz) = s l ( t1) sz(t2) where @ denotes modulo 2 addition. The maximum off-peak auto-correlation for the quasi m-arrays (L1, Lz} while the maximum cross-correlation magon the choice of the m-sequences. This is one reason for the introduction of Gold code arrays in [l]. The maximum off-peak autoand cross-correlation magnitude for the Gold code arraysisgiven by@,,, = max&.(&+l) , L z . ( m t 1 ) 1 . THE NEW CONSTRUCTION In this paper four-phase quasi orthogonal arrays are introduced. The method used for the construction of these arrays from fourphase sequences is similar t o the method used in [l]. BoztaS, Hammons and Kumar [4] constructed families of fourphase linear recurring sequences with near optimum correlation properties. These sequences are used here to construct new families of four-phase quasi-orthogonal code arrays. The reader IS referred to [4] for a tabulation of generating polynomials (hence recursion coefficients) for these sequences. The sequences that are used in the construction here are referred to as the famzly A in that paper and are defined as aU the nonzero sequences satisfying a given linear recursion over Z4. Given two four-phase sequences (say sl(t), t = 0,1, . . . , L1 1 and sz ( t ) , t = 0,1,. . . , LZ I, where L, = 2". 1, with n, a positive z = 1,2) belonging to family A the four-phase quasi array a(t1,tz) of size L1 x Lz can be constructed by a(t1,tz) = S l ( t 1 ) 8 sz(t2) (1) Definition 1 The cross-correlatzon between two four a(t1,tz) and b(t1,tz) of the same dimensions L1 x Lz where 0 5 r, 5 L,, and the sums t, +r, are znterpreted modulo L,, for z = 1,2, and w is defined as eZT1f4. Theorem 1 (a) The cross-correlataon between two four-phase quasz-orthogonal arrays satisfies I @ ) a b ( r l , r Z ) 15 6maz(Ll)flmaz(Jh)7 (3) where OmaZ(L) zs the maxzmum off-peak autoand cross-correlation magnitude of the four-phase sequences in famzly A of length L . (b) The auto-correlation of a four-phase quasi-orthogonal array satasfies I @*47-1,7-2) I5 m ~ { ~ l & " L a ) 1 ~ 2 4 " ( L i ) 1 . (4) Proof The proof is straightforward. Denote the two sequences in family A used to generate b(t1, t z ) by si and si, i.e., Substituting this in Wl, td = s:(tl) e s'z(tz) and a(tl,tz) = S l ( t l ) e s2(tz).