Best Known (119−23, 119, s)-Nets in Base 3
(119−23, 119, 640)-Net over F3 — Constructive and digital
Digital (96, 119, 640)-net over F3, using
- t-expansion [i] based on digital (95, 119, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (95, 120, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 30, 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, 30, 160)-net over F81, using
- 1 times m-reduction [i] based on digital (95, 120, 640)-net over F3, using
(119−23, 119, 2062)-Net over F3 — Digital
Digital (96, 119, 2062)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3119, 2062, F3, 23) (dual of [2062, 1943, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3119, 2222, F3, 23) (dual of [2222, 2103, 24]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [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(385, 2188, F3, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(36, 34, F3, 3) (dual of [34, 28, 4]-code or 34-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3119, 2222, F3, 23) (dual of [2222, 2103, 24]-code), using
(119−23, 119, 322240)-Net in Base 3 — Upper bound on s
There is no (96, 119, 322241)-net in base 3, because
- 1 times m-reduction [i] would yield (96, 118, 322241)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 199 672683 800155 365727 062563 210813 406491 699248 582349 455611 > 3118 [i]