Best Known (187−34, 187, s)-Nets in Base 4
(187−34, 187, 1539)-Net over F4 — Constructive and digital
Digital (153, 187, 1539)-net over F4, using
- 41 times duplication [i] based on digital (152, 186, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 62, 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, 62, 513)-net over F64, using
(187−34, 187, 13440)-Net over F4 — Digital
Digital (153, 187, 13440)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4187, 13440, F4, 34) (dual of [13440, 13253, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4187, 16423, F4, 34) (dual of [16423, 16236, 35]-code), using
- 4 times code embedding in larger space [i] based on linear OA(4183, 16419, F4, 34) (dual of [16419, 16236, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- 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(4148, 16384, F4, 29) (dual of [16384, 16236, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(47, 35, F4, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(33) ⊂ Ce(28) [i] based on
- 4 times code embedding in larger space [i] based on linear OA(4183, 16419, F4, 34) (dual of [16419, 16236, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4187, 16423, F4, 34) (dual of [16423, 16236, 35]-code), using
(187−34, 187, large)-Net in Base 4 — Upper bound on s
There is no (153, 187, large)-net in base 4, because
- 32 times m-reduction [i] would yield (153, 155, large)-net in base 4, but