On the equivalence of three reduced rank linear estimators with applications to DS-CDMA

2609 where H is the natural entropy function and R = k=n is the rate. Since we are dealing with the asymptotical case, we normalize by setting i = n!, and we define the function f (!; R; q) by A !n = e nf(!; R; q) : From Lemma 2, we get f (!; R; q) = H(!) + ! ln(q 0 1) 0 (1 0 R) ln q: (6) Note that, for a given A i , there are two solutions for i. Setting A i 1, the two solutions will be the minimum and the maximum weights. These are, of course, also the zeros of f. Let = d 1 =n and = m 1 =n be, respectively, the minimum and maximum normalized weights. Because and are the zeros of f , we get H() + ln(q 0 1) = H() + ln(q 0 1) Lemma 3 (Varshamov–Gilbert): For almost all linear codes, the rate and the normalized minimum distance are related by the following equation: H() + ln(q 0 1) = (1 0 R) ln q: Proof: This follows from equating f (!; R; q) = 0 as in (6). We know from Theorem 1 that if > 3=4, then the code is (2; 2)-separating. Hence we can, by substituting = 3=4 in the Varshamov–Gilbert equation, get rates for which almost any code is (2; 2)-separating asymptotically. The rates such obtained are presented under " Technique I " in Table I. By the Plotkin bound, this gives nothing over small fields. Technique II in the table is an improvement based on Theorem 1, which says that every code with 4 > 3 is (2; 2)-separating. We insert = 4=3 in (7) and get 3 ln(q 0 1) = ln + (1 0) ln(1 0) 0 4 3 ln 4 3 0 1 0 4 3 ln 1 0 4 3 : (8) We have solved this equation numerically for the smallest fields, and the results are given in Table I. Of course, we will always have 0 1 which will bound 3=4 in (8). This results in no real solution of (8) for q 11. Note that, in Table I, the best results are obtained by the " Constructions " for q 5, then by " Technique II " for 7 q 9, and finally by " Technique I " for higher values of q. In …