Best Known (167, 167+37, s)-Nets in Base 3
(167, 167+37, 896)-Net over F3 — Constructive and digital
Digital (167, 204, 896)-net over F3, using
- t-expansion [i] based on digital (166, 204, 896)-net over F3, using
- trace code for nets [i] based on digital (13, 51, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 51, 224)-net over F81, using
(167, 167+37, 4036)-Net over F3 — Digital
Digital (167, 204, 4036)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3204, 4036, F3, 37) (dual of [4036, 3832, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(3204, 6603, F3, 37) (dual of [6603, 6399, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- linear OA(3193, 6562, F3, 37) (dual of [6562, 6369, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- 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(311, 41, F3, 5) (dual of [41, 30, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- (u, u+v)-construction [i] based on
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3204, 6603, F3, 37) (dual of [6603, 6399, 38]-code), using
(167, 167+37, 907735)-Net in Base 3 — Upper bound on s
There is no (167, 204, 907736)-net in base 3, because
- 1 times m-reduction [i] would yield (167, 203, 907736)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 7 171692 796019 948872 924381 891205 645835 524140 467302 366812 434721 617583 253518 622830 513460 652154 885777 > 3203 [i]