Characteristic $m$-sequences

The initial fc-tuple of the characteristic m-sequence associated with a primitive polynomial of degree k over GF(2) is given for 2 < k < 168. Introduction. In this note we take advantage of the list of primitive polynomials over GF{2) published by Stahnke [1] to calculate a table of characteristic «z-sequences. This author [2] has shown how a characteristic «z-sequence may be used to generate a set of cycle representatives for any cyclic code with square-free parity check polynomial. Such cycle sets are important for determining the error-correcting capability of the cyclic code. In [2] cycle set members are formed by adding certain decimations of a characteristic «¡-sequence. This technique is computationally simpler than standard algorithms based on more complicated algebraic operations. Preliminaries. Let F be the binary field with two elements 0, 1. A polynomial f{x) = xk alxk~1 ■ ■ ■ ak G F[x] is called primitive if a root of f{x) in the extension field K = GF{2k) of F generates the cyclic multiplicative group of K. There are ip(2k \)/k primitive polynomials of degree k, where \p is Euler's function. Assume that f{x) is primitive and consider the linear recursion associated with f{x) given by (1) un + k = alun + k_l + • ■ • +akun, « = 0,1,2,.... Primitive polynomials are characterized by the fact that every nonzero solution to (1) over F has minimum period 2k 1. Therefore, all nonzero solutions to (1) are cyclic shifts of one another. Any such solution is called an m-sequence (or PN sequence). There exists a unique «z-sequence u = {u0, z/j, . . . ) so that un = u2n for all «, called the characteristic «z-sequence associated with/(x). Algorithm. The algorithm used to find the characteristic «z-sequence below is easily adapted to finding such sequences over other prime fields. Treat the symbols u0, «j, . . . , uk_1 as unknowns. From recursion (1) formally calculate uk, uk+l, . . . , M2fc-2> reducing each of these terms to a linear combination of the unknowns. Then solve the system of equations (2) «n=M2#i» " = 0, 1, . . . ,fc-l, for the unknowns. The unique nonzero solution will be the characteristic «z-sequence associated with/(x). The following table lists the initial /c-tuple of the characteristic Received July 1, 1974; revised June 9, 1975. AMS (MOS) subject classifications (1970). Primary 12C05, 12C10; Secondary 94A10. Copyright © 1976, American Mathematical Society