Best Known (232−50, 232, s)-Nets in Base 3
(232−50, 232, 688)-Net over F3 — Constructive and digital
Digital (182, 232, 688)-net over F3, using
- t-expansion [i] based on digital (181, 232, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 58, 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, 58, 172)-net over F81, using
(232−50, 232, 1809)-Net over F3 — Digital
Digital (182, 232, 1809)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3232, 1809, F3, 50) (dual of [1809, 1577, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(3232, 2187, F3, 50) (dual of [2187, 1955, 51]-code), using
- an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- discarding factors / shortening the dual code based on linear OA(3232, 2187, F3, 50) (dual of [2187, 1955, 51]-code), using
(232−50, 232, 136208)-Net in Base 3 — Upper bound on s
There is no (182, 232, 136209)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 492 268360 747724 861263 631480 074636 041416 524161 703267 590678 262600 339549 092005 644017 071727 921311 986884 335617 799619 > 3232 [i]