Best Known (109, 109+35, s)-Nets in Base 3
(109, 109+35, 400)-Net over F3 — Constructive and digital
Digital (109, 144, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 36, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
(109, 109+35, 740)-Net over F3 — Digital
Digital (109, 144, 740)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3144, 740, F3, 35) (dual of [740, 596, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 757, F3, 35) (dual of [757, 613, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- linear OA(3136, 729, F3, 35) (dual of [729, 593, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3112, 729, F3, 29) (dual of [729, 617, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to Ce(34) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3144, 757, F3, 35) (dual of [757, 613, 36]-code), using
(109, 109+35, 36995)-Net in Base 3 — Upper bound on s
There is no (109, 144, 36996)-net in base 3, because
- 1 times m-reduction [i] would yield (109, 143, 36996)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 169 185813 383951 780760 924306 798407 421521 350477 934107 401653 699310 858249 > 3143 [i]