QQRQuinquennial Review (UK)
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In particular, when S is the empty set, ([r.sub.N][r.sub.S], [r.sub.Q][r.sub.S]) is associated with the the zero vector in [F.sup.[2.sup.p]].) We call this a QQR code (or a quasi-quadratic residue code).
Conjecture 3 For p [equivalent to] 1 (mod 4), the associated QQR code and its dual satisfy: [C.sub.NQ] [direct sum] = [F.sup.[2.sup.p]], where [direct sum] stands for the direct product (so, in particular, [C.sub.NQ] [intersection] [C.sup.[perpendicular to].sub.NQ] = {0}).
The following well-known result (v) shall be used to estimate the weights of codewords of QQR codes.
Goppa's conjecture implies that the minimum distance of our QQR code with rate R = 1/2 satisfies d < [[delta].sub.0] x [2.sup.p] = .2[2.sup.p], for sufficiently large p.
This, and the fact that R = 1/2 for our QQR codes (with p [equivalent to] 1 (mod 4)), imply [delta] [less than or equal to] [[delta].sub.0] = [h.sup.-1](1/2) [congruent to] 0.187.
Using SAGE, it can be shown that the Riemann hypothesis is not valid for these "extended QQR codes" in general, as the following example illustrates.
We now introduce a new code, constructed similarly to the QQR codes discussed above: