Best Known (226−42, 226, s)-Nets in Base 4
(226−42, 226, 1539)-Net over F4 — Constructive and digital
Digital (184, 226, 1539)-net over F4, using
- 8 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
(226−42, 226, 12769)-Net over F4 — Digital
Digital (184, 226, 12769)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4226, 12769, F4, 42) (dual of [12769, 12543, 43]-code), using
- discarding factors / shortening the dual code based on linear OA(4226, 16420, F4, 42) (dual of [16420, 16194, 43]-code), using
- 1 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
- 1 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(4226, 16420, F4, 42) (dual of [16420, 16194, 43]-code), using
(226−42, 226, large)-Net in Base 4 — Upper bound on s
There is no (184, 226, large)-net in base 4, because
- 40 times m-reduction [i] would yield (184, 186, large)-net in base 4, but