Best Known (184, 184+48, s)-Nets in Base 3
(184, 184+48, 688)-Net over F3 — Constructive and digital
Digital (184, 232, 688)-net over F3, using
- 4 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+48, 2197)-Net over F3 — Digital
Digital (184, 232, 2197)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3232, 2197, F3, 48) (dual of [2197, 1965, 49]-code), using
- discarding factors / shortening the dual code based on linear OA(3232, 2216, F3, 48) (dual of [2216, 1984, 49]-code), using
- construction XX applied to Ce(48) ⊂ Ce(43) ⊂ Ce(42) [i] based on
- linear OA(3225, 2187, F3, 49) (dual of [2187, 1962, 50]-code), using an extension Ce(48) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,48], and designed minimum distance d ≥ |I|+1 = 49 [i]
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(3197, 2187, F3, 43) (dual of [2187, 1990, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- linear OA(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(48) ⊂ Ce(43) ⊂ Ce(42) [i] based on
- discarding factors / shortening the dual code based on linear OA(3232, 2216, F3, 48) (dual of [2216, 1984, 49]-code), using
(184, 184+48, 200656)-Net in Base 3 — Upper bound on s
There is no (184, 232, 200657)-net in base 3, because
- 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]