Best Known (135−28, 135, s)-Nets in Base 3
(135−28, 135, 640)-Net over F3 — Constructive and digital
Digital (107, 135, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (107, 136, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 34, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 34, 160)-net over F81, using
(135−28, 135, 1494)-Net over F3 — Digital
Digital (107, 135, 1494)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3135, 1494, F3, 28) (dual of [1494, 1359, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3135, 2216, F3, 28) (dual of [2216, 2081, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- linear OA(3127, 2187, F3, 28) (dual of [2187, 2060, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3106, 2187, F3, 23) (dual of [2187, 2081, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(38, 29, F3, 4) (dual of [29, 21, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(27) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3135, 2216, F3, 28) (dual of [2216, 2081, 29]-code), using
(135−28, 135, 120556)-Net in Base 3 — Upper bound on s
There is no (107, 135, 120557)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 25786 145096 617923 322230 111943 883897 009890 730897 555942 300836 192497 > 3135 [i]