Best Known (164, 164+44, s)-Nets in Base 3
(164, 164+44, 688)-Net over F3 — Constructive and digital
Digital (164, 208, 688)-net over F3, using
- t-expansion [i] based on digital (163, 208, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 52, 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, 52, 172)-net over F81, using
(164, 164+44, 1816)-Net over F3 — Digital
Digital (164, 208, 1816)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3208, 1816, F3, 44) (dual of [1816, 1608, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(3208, 2205, F3, 44) (dual of [2205, 1997, 45]-code), using
- construction X applied to Ce(43) ⊂ Ce(40) [i] based on
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- 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(34, 18, F3, 2) (dual of [18, 14, 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 Ce(43) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(3208, 2205, F3, 44) (dual of [2205, 1997, 45]-code), using
(164, 164+44, 146799)-Net in Base 3 — Upper bound on s
There is no (164, 208, 146800)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1742 872728 509521 041132 847257 550476 733353 523253 843563 957028 551969 173745 188984 474562 104770 074671 416481 > 3208 [i]