Best Known (192, 192+44, s)-Nets in Base 4
(192, 192+44, 1539)-Net over F4 — Constructive and digital
Digital (192, 236, 1539)-net over F4, using
- 10 times m-reduction [i] based on digital (192, 246, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 82, 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, 82, 513)-net over F64, using
(192, 192+44, 12829)-Net over F4 — Digital
Digital (192, 236, 12829)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4236, 12829, F4, 44) (dual of [12829, 12593, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(4236, 16409, F4, 44) (dual of [16409, 16173, 45]-code), using
- 1 times truncation [i] based on linear OA(4237, 16410, F4, 45) (dual of [16410, 16173, 46]-code), using
- construction X applied to Ce(44) ⊂ Ce(40) [i] based on
- linear OA(4232, 16384, F4, 45) (dual of [16384, 16152, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(4211, 16384, F4, 41) (dual of [16384, 16173, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(44) ⊂ Ce(40) [i] based on
- 1 times truncation [i] based on linear OA(4237, 16410, F4, 45) (dual of [16410, 16173, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(4236, 16409, F4, 44) (dual of [16409, 16173, 45]-code), using
(192, 192+44, large)-Net in Base 4 — Upper bound on s
There is no (192, 236, large)-net in base 4, because
- 42 times m-reduction [i] would yield (192, 194, large)-net in base 4, but