Best Known (170, 216, s)-Nets in Base 3
(170, 216, 688)-Net over F3 — Constructive and digital
Digital (170, 216, 688)-net over F3, using
- t-expansion [i] based on digital (169, 216, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 54, 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, 54, 172)-net over F81, using
(170, 216, 1810)-Net over F3 — Digital
Digital (170, 216, 1810)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3216, 1810, F3, 46) (dual of [1810, 1594, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 2206, F3, 46) (dual of [2206, 1990, 47]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3215, 2205, F3, 46) (dual of [2205, 1990, 47]-code), using
- construction X applied to Ce(45) ⊂ Ce(42) [i] based on
- linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(34, 18, F3, 2) (dual of [18, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(45) ⊂ Ce(42) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3215, 2205, F3, 46) (dual of [2205, 1990, 47]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 2206, F3, 46) (dual of [2206, 1990, 47]-code), using
(170, 216, 142609)-Net in Base 3 — Upper bound on s
There is no (170, 216, 142610)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 11 434593 735804 659213 091591 544078 093456 290000 225185 725304 810643 413273 860529 476424 612101 955129 823429 691017 > 3216 [i]