Best Known (201−41, 201, s)-Nets in Base 3
(201−41, 201, 688)-Net over F3 — Constructive and digital
Digital (160, 201, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (160, 204, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 51, 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, 51, 172)-net over F81, using
(201−41, 201, 2117)-Net over F3 — Digital
Digital (160, 201, 2117)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3201, 2117, F3, 41) (dual of [2117, 1916, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(3201, 2226, F3, 41) (dual of [2226, 2025, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- linear OA(3190, 2187, F3, 41) (dual of [2187, 1997, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(311, 39, F3, 5) (dual of [39, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(37, 14, F3, 5) (dual of [14, 7, 6]-code), using
- extended quadratic residue code Qe(14,3) [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(311, 40, F3, 5) (dual of [40, 29, 6]-code), using
- construction X applied to Ce(40) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3201, 2226, F3, 41) (dual of [2226, 2025, 42]-code), using
(201−41, 201, 245162)-Net in Base 3 — Upper bound on s
There is no (160, 201, 245163)-net in base 3, because
- 1 times m-reduction [i] would yield (160, 200, 245163)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 265622 373812 434123 817249 643473 399500 687430 690331 183255 713435 186294 383355 376025 271073 609775 454649 > 3200 [i]