Best Known (89, 111, s)-Nets in Base 3
(89, 111, 640)-Net over F3 — Constructive and digital
Digital (89, 111, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (89, 112, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 28, 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, 28, 160)-net over F81, using
(89, 111, 1729)-Net over F3 — Digital
Digital (89, 111, 1729)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3111, 1729, F3, 22) (dual of [1729, 1618, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3111, 2227, F3, 22) (dual of [2227, 2116, 23]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3109, 2225, F3, 22) (dual of [2225, 2116, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(399, 2187, F3, 22) (dual of [2187, 2088, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(371, 2187, F3, 16) (dual of [2187, 2116, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(310, 38, F3, 5) (dual of [38, 28, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 39, F3, 5) (dual of [39, 29, 6]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3109, 2225, F3, 22) (dual of [2225, 2116, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3111, 2227, F3, 22) (dual of [2227, 2116, 23]-code), using
(89, 111, 160155)-Net in Base 3 — Upper bound on s
There is no (89, 111, 160156)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 91298 315504 003126 122399 573385 003337 689547 176534 353233 > 3111 [i]