Best Known (169−35, 169, s)-Nets in Base 3
(169−35, 169, 688)-Net over F3 — Constructive and digital
Digital (134, 169, 688)-net over F3, using
- 31 times duplication [i] based on digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 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, 42, 172)-net over F81, using
(169−35, 169, 1737)-Net over F3 — Digital
Digital (134, 169, 1737)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3169, 1737, F3, 35) (dual of [1737, 1568, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3169, 2215, F3, 35) (dual of [2215, 2046, 36]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3168, 2214, F3, 35) (dual of [2214, 2046, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(36, 27, F3, 3) (dual of [27, 21, 4]-code or 27-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
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3168, 2214, F3, 35) (dual of [2214, 2046, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3169, 2215, F3, 35) (dual of [2215, 2046, 36]-code), using
(169−35, 169, 186190)-Net in Base 3 — Upper bound on s
There is no (134, 169, 186191)-net in base 3, because
- 1 times m-reduction [i] would yield (134, 168, 186191)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 143 348537 328064 693472 479387 895247 706373 223334 714618 219806 706110 265131 739508 223615 > 3168 [i]