Best Known (164, 164+38, s)-Nets in Base 3
(164, 164+38, 698)-Net over F3 — Constructive and digital
Digital (164, 202, 698)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 22, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- digital (142, 180, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 45, 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, 45, 172)-net over F81, using
- digital (3, 22, 10)-net over F3, using
(164, 164+38, 3285)-Net over F3 — Digital
Digital (164, 202, 3285)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3202, 3285, F3, 2, 38) (dual of [(3285, 2), 6368, 39]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3202, 6570, F3, 38) (dual of [6570, 6368, 39]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3201, 6569, F3, 38) (dual of [6569, 6368, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(3201, 6561, F3, 38) (dual of [6561, 6360, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3193, 6561, F3, 37) (dual of [6561, 6368, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3201, 6569, F3, 38) (dual of [6569, 6368, 39]-code), using
- OOA 2-folding [i] based on linear OA(3202, 6570, F3, 38) (dual of [6570, 6368, 39]-code), using
(164, 164+38, 468511)-Net in Base 3 — Upper bound on s
There is no (164, 202, 468512)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2 390533 164066 977630 026618 065068 148960 757500 012431 709788 567864 334481 045470 996868 304475 076865 399937 > 3202 [i]