Best Known (133, 133+31, s)-Nets in Base 3
(133, 133+31, 688)-Net over F3 — Constructive and digital
Digital (133, 164, 688)-net over F3, using
- 4 times m-reduction [i] based on digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
(133, 133+31, 3227)-Net over F3 — Digital
Digital (133, 164, 3227)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3164, 3227, F3, 2, 31) (dual of [(3227, 2), 6290, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3164, 3287, F3, 2, 31) (dual of [(3287, 2), 6410, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3164, 6574, F3, 31) (dual of [6574, 6410, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3161, 6561, F3, 31) (dual of [6561, 6400, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(3164, 6574, F3, 31) (dual of [6574, 6410, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(3164, 3287, F3, 2, 31) (dual of [(3287, 2), 6410, 32]-NRT-code), using
(133, 133+31, 491407)-Net in Base 3 — Upper bound on s
There is no (133, 164, 491408)-net in base 3, because
- 1 times m-reduction [i] would yield (133, 163, 491408)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 589896 576835 923560 580889 357783 090536 536661 938092 904985 830213 717185 976136 724161 > 3163 [i]