Best Known (184, 184+49, s)-Nets in Base 3
(184, 184+49, 688)-Net over F3 — Constructive and digital
Digital (184, 233, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (184, 236, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 59, 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, 59, 172)-net over F81, using
(184, 184+49, 2037)-Net over F3 — Digital
Digital (184, 233, 2037)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3233, 2037, F3, 49) (dual of [2037, 1804, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(3233, 2216, F3, 49) (dual of [2216, 1983, 50]-code), using
- construction X applied to C([0,24]) ⊂ C([0,21]) [i] based on
- linear OA(3225, 2188, F3, 49) (dual of [2188, 1963, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- linear OA(3197, 2188, F3, 43) (dual of [2188, 1991, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to C([0,24]) ⊂ C([0,21]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3233, 2216, F3, 49) (dual of [2216, 1983, 50]-code), using
(184, 184+49, 200656)-Net in Base 3 — Upper bound on s
There is no (184, 233, 200657)-net in base 3, because
- 1 times m-reduction [i] would yield (184, 232, 200657)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 492 196308 197855 048349 928170 083941 541062 084657 420482 177638 899975 131868 456322 700103 008975 313850 805455 710465 803489 > 3232 [i]