Best Known (194−36, 194, s)-Nets in Base 4
(194−36, 194, 1539)-Net over F4 — Constructive and digital
Digital (158, 194, 1539)-net over F4, using
- 1 times m-reduction [i] based on digital (158, 195, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 65, 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, 65, 513)-net over F64, using
(194−36, 194, 11774)-Net over F4 — Digital
Digital (158, 194, 11774)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4194, 11774, F4, 36) (dual of [11774, 11580, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(4194, 16409, F4, 36) (dual of [16409, 16215, 37]-code), using
- 1 times truncation [i] based on linear OA(4195, 16410, F4, 37) (dual of [16410, 16215, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(32) [i] based on
- linear OA(4190, 16384, F4, 37) (dual of [16384, 16194, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4169, 16384, F4, 33) (dual of [16384, 16215, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(36) ⊂ Ce(32) [i] based on
- 1 times truncation [i] based on linear OA(4195, 16410, F4, 37) (dual of [16410, 16215, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4194, 16409, F4, 36) (dual of [16409, 16215, 37]-code), using
(194−36, 194, 7760269)-Net in Base 4 — Upper bound on s
There is no (158, 194, 7760270)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 630 433297 066923 164757 579079 404852 373119 716312 447901 956681 635615 288403 985992 013172 042952 324122 060853 193875 531602 524112 > 4194 [i]