Gold sequence

and Information
theory
vol. MIL-9, pp. 56-69.
with Applications to
for high-velocity
Theor2,
1 Since the preparation of this paper, we have noted the work of Kasami[61 where similar results have been obtained. * This theorem translated in coding theory reads as follows. ThClTl3WZ Let n be odd. Then the (2” - 1, 2%) cyclic code generated by the polynomial z via linear recursion is a tri-weight code with nonzero weights ~P&?~~~2tn-$)*. H erefr(z) is a primitive polynomial over CF(B*) with root a and hi +; (2) is an irreducible polynomial with azl+’ m a root. 1 is prime to n, (1, n) = 1. Furthermore. the weight distribution for these codes is as follows. N(w) is the number of code words of weight o in the code. w N(w)
n-1
We shall represent where (Y is a primitive trace of GF(2”), i.e.,
= E-’lrn IG(f)12 f-’df.
With the same assumptions used above, Vz is also approximated
Maximal Recursive Sequences with 3-Valued Recursive Cross-Correlation Functions
I. INTRODUCTION Maximal linear sequences have communication systems where a ministic component is added to the detection, or spectrum spreading found extensive applications in random-like but in fact deterbasic signal for synchronization, purposes.[il One of the most
(12)
by
!I’ @) = c /32i for ,f3E GF(2r). i=0
T is a homomorphism of the additive group of GF(2”) onto the group (0, 1 ) and for such maps it is known131 .[*I that h is a maximal recursive sequence and that h P V(f) where f is the minimal polynomial of 01 and V(f) is the space of linear sequences given by (h Ih = gf-i; degree g < degree f 1. As a notational convenience we shall want to represent sequences as real valued sequences of plus-minus ones. For this purpose we define S = poT where ‘ p is the mapping such that ~(0) = 1 and ~p(1) = - 1. The cross-correlation sequence of two such real valued sequences a and b is then defined as 2”-2
154 of a relatively small bandwidth, then VI can be approximated
IEEE
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INFORMATION
THEORY,
JANUARY
1968
by
) I2lfo I df’ . I- P(f) I2f02df Vl = E-z [--03 IW’
(7)
The above relationship is exactly the expression obtained by Kelly and Wishnerl21 after defining the generalized ambiguity function as
x2(7, PI = E-’jm s(4P”“s*W -cc
In t,his case
=E-1 _: WP-““G*(fIP) s
(10)
82 = j 0 1 -m /x2(7, P)l” dT @ (11)
(9)
important and well-known properties of such sequences is their ideal autocorrelat,ion propertyI*]: if a maximal linear sequence is compared with any proper phase shift of itself over its period, the number of terms for which they agree minus the number of terms for which they disagree is always minus one. On the other hand, little has been known concerning the cross-correlation function of two different maximal sequences of the same period. In this correspondence we describe a class of pairs of maximal linear sequences for which we can determine precisely the cross-correlation function. These cross-correlation functions have three values and furthermore have the remarkable property that they themselves are maximal linear sequences when two of the three correlation values are properly identified. In coding theory terms, we have a class of cyclic codes having three nonzero weights. Their distribution is the same as that of the Bose-Chaudhuri-Hocquenhem code of the same dimension. II. NOTATIONS AND CONVENTIONS the maximal sequence h as h(i) = T(a-“) 2% - 1 root of unity in GF(2”) and T is the
The If a = d = 0 GF(2”)
2”--’ Y-’ + 2’ “ -“ ‘ 2
y-1 _ 2h-1)/2
(2” (2” (2” -
1)(2”-’ + 1) “ ~3’ /2 1) (2,-, _ 2’
1)(2”-2 + 2+-3)/2
>
).
algebraic characterization of these codes via theor weights are the following. (a). a, = Tr czi f Tr dz@‘ +*)i. c. d, 6 GF(2 c # 0 Tr cd@‘ +1) = 0, then w(a) = 2”-1. If Tr cd ‘ 21+‘ )-1 = 1 and there exist 60 el+l)-l and Tr (B + /321+1)= 0, then such that fl*l + 82-I f 1 = cd<
lwi7.
Thus 6’ (a, b)(7) = number of agreements - number of disagreements when sequences a and b are aligned in relative position T. III. We now prove Theorem’ 92 Let a and 6 be maximal linear sequences given by a(i) = S( a-i) STATE~WENTAND PROOF OF RESULT the following theorem.
Jmm-y 1965.
Radar.
~1 E. J. Kelly and R. P. Wishner. “Matched-filter accelerating targets,” IEEE Trans. Military Electronics, [al P. M. Woodward. Probability London: Pergamon, 1953.