Best Known (211, 211+47, s)-Nets in Base 4
(211, 211+47, 1566)-Net over F4 — Constructive and digital
Digital (211, 258, 1566)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (10, 33, 27)-net over F4, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 10 and N(F) ≥ 27, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- digital (178, 225, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 75, 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, 75, 513)-net over F64, using
- digital (10, 33, 27)-net over F4, using
(211, 211+47, 16088)-Net over F4 — Digital
Digital (211, 258, 16088)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4258, 16088, F4, 47) (dual of [16088, 15830, 48]-code), using
- discarding factors / shortening the dual code based on linear OA(4258, 16418, F4, 47) (dual of [16418, 16160, 48]-code), using
- construction X applied to C([0,24]) ⊂ C([0,21]) [i] based on
- linear OA(4253, 16385, F4, 49) (dual of [16385, 16132, 50]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,24], and minimum distance d ≥ |{−24,−23,…,24}|+1 = 50 (BCH-bound) [i]
- linear OA(4225, 16385, F4, 43) (dual of [16385, 16160, 44]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,21], and minimum distance d ≥ |{−21,−20,…,21}|+1 = 44 (BCH-bound) [i]
- linear OA(45, 33, F4, 3) (dual of [33, 28, 4]-code or 33-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
- construction X applied to C([0,24]) ⊂ C([0,21]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4258, 16418, F4, 47) (dual of [16418, 16160, 48]-code), using
(211, 211+47, large)-Net in Base 4 — Upper bound on s
There is no (211, 258, large)-net in base 4, because
- 45 times m-reduction [i] would yield (211, 213, large)-net in base 4, but