Best Known (107−21, 107, s)-Nets in Base 3
(107−21, 107, 640)-Net over F3 — Constructive and digital
Digital (86, 107, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (86, 108, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 27, 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, 27, 160)-net over F81, using
(107−21, 107, 1802)-Net over F3 — Digital
Digital (86, 107, 1802)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3107, 1802, F3, 21) (dual of [1802, 1695, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3107, 2216, F3, 21) (dual of [2216, 2109, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- 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(371, 2188, F3, 15) (dual of [2188, 2117, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [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 C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3107, 2216, F3, 21) (dual of [2216, 2109, 22]-code), using
(107−21, 107, 258473)-Net in Base 3 — Upper bound on s
There is no (86, 107, 258474)-net in base 3, because
- 1 times m-reduction [i] would yield (86, 106, 258474)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 375 720747 336596 403728 734437 134574 119319 200995 726101 > 3106 [i]