Best Known (94, 94+25, s)-Nets in Base 3
(94, 94+25, 600)-Net over F3 — Constructive and digital
Digital (94, 119, 600)-net over F3, using
- 1 times m-reduction [i] based on digital (94, 120, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 30, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- trace code for nets [i] based on digital (4, 30, 150)-net over F81, using
(94, 94+25, 1301)-Net over F3 — Digital
Digital (94, 119, 1301)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3119, 1301, F3, 25) (dual of [1301, 1182, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3119, 2208, F3, 25) (dual of [2208, 2089, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3118, 2207, F3, 25) (dual of [2207, 2089, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- linear OA(3113, 2188, F3, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(399, 2188, F3, 21) (dual of [2188, 2089, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(35, 19, F3, 3) (dual of [19, 14, 4]-code or 19-cap in PG(4,3)), using
- construction X applied to C([0,12]) ⊂ C([0,10]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3118, 2207, F3, 25) (dual of [2207, 2089, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3119, 2208, F3, 25) (dual of [2208, 2089, 26]-code), using
(94, 94+25, 130012)-Net in Base 3 — Upper bound on s
There is no (94, 119, 130013)-net in base 3, because
- 1 times m-reduction [i] would yield (94, 118, 130013)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 199 675727 586657 794840 598046 474770 483243 139795 047046 627569 > 3118 [i]