Best Known (249−47, 249, s)-Nets in Base 3
(249−47, 249, 896)-Net over F3 — Constructive and digital
Digital (202, 249, 896)-net over F3, using
- 31 times duplication [i] based on digital (201, 248, 896)-net over F3, using
- t-expansion [i] based on digital (199, 248, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 62, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 62, 224)-net over F81, using
- t-expansion [i] based on digital (199, 248, 896)-net over F3, using
(249−47, 249, 3712)-Net over F3 — Digital
Digital (202, 249, 3712)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3249, 3712, F3, 47) (dual of [3712, 3463, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(3249, 6561, F3, 47) (dual of [6561, 6312, 48]-code), using
- an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- discarding factors / shortening the dual code based on linear OA(3249, 6561, F3, 47) (dual of [6561, 6312, 48]-code), using
(249−47, 249, 657692)-Net in Base 3 — Upper bound on s
There is no (202, 249, 657693)-net in base 3, because
- 1 times m-reduction [i] would yield (202, 248, 657693)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 21187 086130 093863 176501 766493 972572 308039 569770 802329 106236 450972 561533 217187 238677 421634 809211 869168 871101 870399 104123 > 3248 [i]