Best Known (146, 179, s)-Nets in Base 3
(146, 179, 698)-Net over F3 — Constructive and digital
Digital (146, 179, 698)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 19, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- digital (127, 160, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 40, 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, 40, 172)-net over F81, using
- digital (3, 19, 10)-net over F3, using
(146, 179, 3379)-Net over F3 — Digital
Digital (146, 179, 3379)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3179, 3379, F3, 33) (dual of [3379, 3200, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3179, 6580, F3, 33) (dual of [6580, 6401, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3178, 6579, F3, 33) (dual of [6579, 6401, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- linear OA(3177, 6562, F3, 33) (dual of [6562, 6385, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3161, 6562, F3, 31) (dual of [6562, 6401, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3178, 6579, F3, 33) (dual of [6579, 6401, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3179, 6580, F3, 33) (dual of [6580, 6401, 34]-code), using
(146, 179, 690991)-Net in Base 3 — Upper bound on s
There is no (146, 179, 690992)-net in base 3, because
- 1 times m-reduction [i] would yield (146, 178, 690992)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 8 464172 644556 541063 088095 234361 619375 075775 634570 675511 836346 824848 866625 757786 655745 > 3178 [i]