Best Known (141−25, 141, s)-Nets in Base 3
(141−25, 141, 695)-Net over F3 — Constructive and digital
Digital (116, 141, 695)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (103, 128, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 32, 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, 32, 172)-net over F81, using
- digital (1, 13, 7)-net over F3, using
(141−25, 141, 3760)-Net over F3 — Digital
Digital (116, 141, 3760)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3141, 3760, F3, 25) (dual of [3760, 3619, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3141, 6575, F3, 25) (dual of [6575, 6434, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([1,12]) [i] based on
- linear OA(3129, 6562, F3, 25) (dual of [6562, 6433, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(3128, 6562, F3, 12) (dual of [6562, 6434, 13]-code), using the narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(312, 13, F3, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
- dual of repetition code with length 13 [i]
- construction X applied to C([0,12]) ⊂ C([1,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3141, 6575, F3, 25) (dual of [6575, 6434, 26]-code), using
(141−25, 141, 974408)-Net in Base 3 — Upper bound on s
There is no (116, 141, 974409)-net in base 3, because
- 1 times m-reduction [i] would yield (116, 140, 974409)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6 265849 065603 279790 526397 114217 642889 493993 786099 612550 531790 004657 > 3140 [i]