Best Known (130−37, 130, s)-Nets in Base 9
(130−37, 130, 774)-Net over F9 — Constructive and digital
Digital (93, 130, 774)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (6, 24, 34)-net over F9, using
- net from sequence [i] based on digital (6, 33)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 6 and N(F) ≥ 34, using
- net from sequence [i] based on digital (6, 33)-sequence over F9, using
- digital (69, 106, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 53, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 53, 370)-net over F81, using
- digital (6, 24, 34)-net over F9, using
(130−37, 130, 5697)-Net over F9 — Digital
Digital (93, 130, 5697)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9130, 5697, F9, 37) (dual of [5697, 5567, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(9130, 6566, F9, 37) (dual of [6566, 6436, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(9129, 6561, F9, 37) (dual of [6561, 6432, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(9125, 6561, F9, 35) (dual of [6561, 6436, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(91, 5, F9, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(9130, 6566, F9, 37) (dual of [6566, 6436, 38]-code), using
(130−37, 130, 6512943)-Net in Base 9 — Upper bound on s
There is no (93, 130, 6512944)-net in base 9, because
- 1 times m-reduction [i] would yield (93, 129, 6512944)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 1251 076425 847928 341712 156910 653980 408695 636622 291399 485363 064673 974135 790403 016698 870875 894315 842797 312915 278655 602492 342529 > 9129 [i]