Best Known (163−30, 163, s)-Nets in Base 3
(163−30, 163, 692)-Net over F3 — Constructive and digital
Digital (133, 163, 692)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 15, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (118, 148, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 37, 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, 37, 172)-net over F81, using
- digital (0, 15, 4)-net over F3, using
(163−30, 163, 3289)-Net over F3 — Digital
Digital (133, 163, 3289)-net over F3, using
- 31 times duplication [i] based on digital (132, 162, 3289)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3162, 3289, F3, 2, 30) (dual of [(3289, 2), 6416, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3162, 6578, F3, 30) (dual of [6578, 6416, 31]-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(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 Ce(30) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(3162, 6578, F3, 30) (dual of [6578, 6416, 31]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3162, 3289, F3, 2, 30) (dual of [(3289, 2), 6416, 31]-NRT-code), using
(163−30, 163, 491407)-Net in Base 3 — Upper bound on s
There is no (133, 163, 491408)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 589896 576835 923560 580889 357783 090536 536661 938092 904985 830213 717185 976136 724161 > 3163 [i]