Best Known (132, 161, s)-Nets in Base 3
(132, 161, 698)-Net over F3 — Constructive and digital
Digital (132, 161, 698)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 17, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-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 (3, 17, 10)-net over F3, using
(132, 161, 3645)-Net over F3 — Digital
Digital (132, 161, 3645)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3161, 3645, F3, 29) (dual of [3645, 3484, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3161, 6594, F3, 29) (dual of [6594, 6433, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3153, 6561, F3, 29) (dual of [6561, 6408, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3129, 6561, F3, 25) (dual of [6561, 6432, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3121, 6561, F3, 23) (dual of [6561, 6440, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(36, 31, F3, 3) (dual of [31, 25, 4]-code or 31-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3161, 6594, F3, 29) (dual of [6594, 6433, 30]-code), using
(132, 161, 857502)-Net in Base 3 — Upper bound on s
There is no (132, 161, 857503)-net in base 3, because
- 1 times m-reduction [i] would yield (132, 160, 857503)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 21847 733071 423695 486236 760879 883943 303262 668907 709104 666613 182843 126306 604581 > 3160 [i]