Best Known (227−42, 227, s)-Nets in Base 4
(227−42, 227, 1539)-Net over F4 — Constructive and digital
Digital (185, 227, 1539)-net over F4, using
- t-expansion [i] based on digital (184, 227, 1539)-net over F4, using
- 7 times m-reduction [i] based on digital (184, 234, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 78, 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, 78, 513)-net over F64, using
- 7 times m-reduction [i] based on digital (184, 234, 1539)-net over F4, using
(227−42, 227, 13221)-Net over F4 — Digital
Digital (185, 227, 13221)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4227, 13221, F4, 42) (dual of [13221, 12994, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4227, 16421, F4, 42) (dual of [16421, 16194, 43]-code), using
- 2 times code embedding in larger space [i] based on linear OA(4225, 16419, F4, 42) (dual of [16419, 16194, 43]-code), using
- construction X applied to Ce(41) ⊂ Ce(36) [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(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(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(41) ⊂ Ce(36) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(4225, 16419, F4, 42) (dual of [16419, 16194, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4227, 16421, F4, 42) (dual of [16421, 16194, 43]-code), using
(227−42, 227, large)-Net in Base 4 — Upper bound on s
There is no (185, 227, large)-net in base 4, because
- 40 times m-reduction [i] would yield (185, 187, large)-net in base 4, but