Best Known (135−29, 135, s)-Nets in Base 3
(135−29, 135, 600)-Net over F3 — Constructive and digital
Digital (106, 135, 600)-net over F3, using
- 1 times m-reduction [i] based on digital (106, 136, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 34, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- trace code for nets [i] based on digital (4, 34, 150)-net over F81, using
(135−29, 135, 1250)-Net over F3 — Digital
Digital (106, 135, 1250)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3135, 1250, F3, 29) (dual of [1250, 1115, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3135, 2195, F3, 29) (dual of [2195, 2060, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3134, 2194, F3, 29) (dual of [2194, 2060, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3134, 2187, F3, 29) (dual of [2187, 2053, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(30, 7, F3, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3134, 2194, F3, 29) (dual of [2194, 2060, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3135, 2195, F3, 29) (dual of [2195, 2060, 30]-code), using
(135−29, 135, 111456)-Net in Base 3 — Upper bound on s
There is no (106, 135, 111457)-net in base 3, because
- 1 times m-reduction [i] would yield (106, 134, 111457)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8595 098103 807232 974070 038664 136785 093663 976835 378769 057298 215737 > 3134 [i]