Best Known (192, 234, s)-Nets in Base 4
(192, 234, 1553)-Net over F4 — Constructive and digital
Digital (192, 234, 1553)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 24, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (168, 210, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 70, 513)-net over F64, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- the Hermitian function field over F64 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 28 and N(F) ≥ 513, using
- net from sequence [i] based on digital (28, 512)-sequence over F64, using
- trace code for nets [i] based on digital (28, 70, 513)-net over F64, using
- digital (3, 24, 14)-net over F4, using
(192, 234, 16443)-Net over F4 — Digital
Digital (192, 234, 16443)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4234, 16443, F4, 42) (dual of [16443, 16209, 43]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4232, 16440, F4, 42) (dual of [16440, 16208, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(33) [i] based on
- linear OA(4218, 16384, F4, 42) (dual of [16384, 16166, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(4176, 16384, F4, 34) (dual of [16384, 16208, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(414, 56, F4, 7) (dual of [56, 42, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- a “GraXX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(414, 65, F4, 7) (dual of [65, 51, 8]-code), using
- construction X applied to Ce(41) ⊂ Ce(33) [i] based on
- linear OA(4232, 16441, F4, 40) (dual of [16441, 16209, 41]-code), using Gilbert–Varšamov bound and bm = 4232 > Vbs−1(k−1) = 50011 360254 408330 512553 658232 128297 102883 411717 370148 276172 752057 393406 531559 282848 176511 436705 810054 307638 635487 733972 508690 163093 085621 [i]
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(4232, 16440, F4, 42) (dual of [16440, 16208, 43]-code), using
- construction X with Varšamov bound [i] based on
(192, 234, large)-Net in Base 4 — Upper bound on s
There is no (192, 234, large)-net in base 4, because
- 40 times m-reduction [i] would yield (192, 194, large)-net in base 4, but