Best Known (110, 110+29, s)-Nets in Base 3
(110, 110+29, 640)-Net over F3 — Constructive and digital
Digital (110, 139, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (110, 140, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 35, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 35, 160)-net over F81, using
(110, 110+29, 1475)-Net over F3 — Digital
Digital (110, 139, 1475)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3139, 1475, F3, 29) (dual of [1475, 1336, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3139, 2207, F3, 29) (dual of [2207, 2068, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3113, 2187, F3, 25) (dual of [2187, 2074, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(28) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3139, 2207, F3, 29) (dual of [2207, 2068, 30]-code), using
(110, 110+29, 152560)-Net in Base 3 — Upper bound on s
There is no (110, 139, 152561)-net in base 3, because
- 1 times m-reduction [i] would yield (110, 138, 152561)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 696239 955921 931079 939527 157268 840945 865762 591638 704306 125720 521177 > 3138 [i]