Best Known (47−13, 47, s)-Nets in Base 4
(47−13, 47, 240)-Net over F4 — Constructive and digital
Digital (34, 47, 240)-net over F4, using
- t-expansion [i] based on digital (33, 47, 240)-net over F4, using
- 1 times m-reduction [i] based on digital (33, 48, 240)-net over F4, using
- trace code for nets [i] based on digital (1, 16, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 16, 80)-net over F64, using
- 1 times m-reduction [i] based on digital (33, 48, 240)-net over F4, using
(47−13, 47, 531)-Net over F4 — Digital
Digital (34, 47, 531)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(447, 531, F4, 13) (dual of [531, 484, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(447, 1030, F4, 13) (dual of [1030, 983, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(446, 1024, F4, 13) (dual of [1024, 978, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(441, 1024, F4, 11) (dual of [1024, 983, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(41, 6, F4, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(447, 1030, F4, 13) (dual of [1030, 983, 14]-code), using
(47−13, 47, 41195)-Net in Base 4 — Upper bound on s
There is no (34, 47, 41196)-net in base 4, because
- 1 times m-reduction [i] would yield (34, 46, 41196)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 4952 334133 493359 425116 801974 > 446 [i]