Best Known (110, 110+31, s)-Nets in Base 3
(110, 110+31, 600)-Net over F3 — Constructive and digital
Digital (110, 141, 600)-net over F3, using
- 31 times duplication [i] based on digital (109, 140, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 35, 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, 35, 150)-net over F81, using
(110, 110+31, 1147)-Net over F3 — Digital
Digital (110, 141, 1147)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3141, 1147, F3, 31) (dual of [1147, 1006, 32]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(3141, 2187, F3, 31) (dual of [2187, 2046, 32]-code), using
(110, 110+31, 91158)-Net in Base 3 — Upper bound on s
There is no (110, 141, 91159)-net in base 3, because
- 1 times m-reduction [i] would yield (110, 140, 91159)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 265845 171802 550342 917512 479503 903558 090980 608024 586196 121947 080771 > 3140 [i]