Best Known (154, 154+35, s)-Nets in Base 4
(154, 154+35, 1539)-Net over F4 — Constructive and digital
Digital (154, 189, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 63, 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
(154, 154+35, 11781)-Net over F4 — Digital
Digital (154, 189, 11781)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4189, 11781, F4, 35) (dual of [11781, 11592, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(4189, 16412, F4, 35) (dual of [16412, 16223, 36]-code), using
- construction XX applied to Ce(34) ⊂ Ce(30) ⊂ Ce(29) [i] based on
- linear OA(4183, 16384, F4, 35) (dual of [16384, 16201, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(4162, 16384, F4, 31) (dual of [16384, 16222, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4155, 16384, F4, 30) (dual of [16384, 16229, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(34) ⊂ Ce(30) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(4189, 16412, F4, 35) (dual of [16412, 16223, 36]-code), using
(154, 154+35, large)-Net in Base 4 — Upper bound on s
There is no (154, 189, large)-net in base 4, because
- 33 times m-reduction [i] would yield (154, 156, large)-net in base 4, but