e roots of the same irreducible poly- nomial have the same trace value; hence, trace values can be computed by (5.126) and associated with cyclotomic cosets. This is illustrated in Table 5.9 for this example. The following theorem provides a convenient representation for certain LFSR sequences having irreducible characteristic polynomials. Here, and in discussions to follow, the superscripts and subscripts on the trace function will be omitted when their values are evident. THEOREM 5.6. Let GF(qd) be the smallest field containing the element a, and let ma(z) be the degree d minimum polynomial of a over GF(q). Then the sequence {Trq(a")}, n being the sequence index, has characteristic polynomial ma(z). Proof. Consider a d-cell Galois LFSR producing an output sequence hav- ing characteristic polynomial ma(z). Combining the representation (5.22) for the remainder polynomial in terms of memory contents, with the field ele- ment interpretation of (5.112) gives (5.127) where B=[P(a)]-1 (5.128) with a and B in GF(q). Applying the trace function with Property T-3 to both sides of (5.127) and noting that the cell sequences are elements of GF(q), gives (5.129) Pseudonoise Generators Corollary 5.1 indicates that the L memory cell sequences {r()}, i = 1, ***, , L, all have the same characteristic polynomial ma(z); therefore, the linear com- bination of those sequences specified by (5.129), namely the sequence {Tr(a")}, has the same characteristic polynomial. The following corollary provides an explicit and compact mathematical representation of the elements of an m-sequence, which will be used in sev- eral forthcoming analyses. COROLLARY 5.2. If {bn} is an m-sequence over GF(q) with the minimum polynomial ma(z) as its characteristic polynomial, then there exists a non- zero element Y in GF(qd) such that {b,} = {Tr,(ya")}. (5.130) Proof. By Theorem 5.6, the same recursion which generates {bn} must also produce {Tr(a")}, as well as any shift thereof. Since {bn} is an m-sequence, its characteristic polynomial ma(z) is primitive