Best Known (110−22, 110, s)-Nets in Base 3
(110−22, 110, 640)-Net over F3 — Constructive and digital
Digital (88, 110, 640)-net over F3, using
- 32 times duplication [i] based on digital (86, 108, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 27, 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, 27, 160)-net over F81, using
(110−22, 110, 1636)-Net over F3 — Digital
Digital (88, 110, 1636)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3110, 1636, F3, 22) (dual of [1636, 1526, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3110, 2199, F3, 22) (dual of [2199, 2089, 23]-code), using
- (u, u+v)-construction [i] based on
- linear OA(311, 12, F3, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
- dual of repetition code with length 12 [i]
- 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(311, 12, F3, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,3)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(3110, 2199, F3, 22) (dual of [2199, 2089, 23]-code), using
(110−22, 110, 144932)-Net in Base 3 — Upper bound on s
There is no (88, 110, 144933)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 30434 096236 137863 561372 065668 783630 865215 071584 305547 > 3110 [i]