Best Known (165−30, 165, s)-Nets in Base 3
(165−30, 165, 696)-Net over F3 — Constructive and digital
Digital (135, 165, 696)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 17, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (118, 148, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 37, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 37, 172)-net over F81, using
- digital (2, 17, 8)-net over F3, using
(165−30, 165, 3494)-Net over F3 — Digital
Digital (135, 165, 3494)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3165, 3494, F3, 30) (dual of [3494, 3329, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3165, 6582, F3, 30) (dual of [6582, 6417, 31]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(3161, 6562, F3, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3145, 6562, F3, 27) (dual of [6562, 6417, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(34, 20, F3, 2) (dual of [20, 16, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3165, 6582, F3, 30) (dual of [6582, 6417, 31]-code), using
(165−30, 165, 568930)-Net in Base 3 — Upper bound on s
There is no (135, 165, 568931)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 5 308970 641568 126407 810069 598912 110838 722508 667403 513052 087331 534645 837568 795251 > 3165 [i]