Best Known (133, 133+29, s)-Nets in Base 3
(133, 133+29, 700)-Net over F3 — Constructive and digital
Digital (133, 162, 700)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 18, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- digital (115, 144, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 36, 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, 36, 172)-net over F81, using
- digital (4, 18, 12)-net over F3, using
(133, 133+29, 3798)-Net over F3 — Digital
Digital (133, 162, 3798)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3162, 3798, F3, 29) (dual of [3798, 3636, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3162, 6579, F3, 29) (dual of [6579, 6417, 30]-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(31, 17, F3, 1) (dual of [17, 16, 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
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3162, 6579, F3, 29) (dual of [6579, 6417, 30]-code), using
(133, 133+29, 927504)-Net in Base 3 — Upper bound on s
There is no (133, 162, 927505)-net in base 3, because
- 1 times m-reduction [i] would yield (133, 161, 927505)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 65543 211343 617923 207642 430041 678730 283535 505369 179076 496409 279379 872546 626329 > 3161 [i]