Best Known (209, 258, s)-Nets in Base 4
(209, 258, 1553)-Net over F4 — Constructive and digital
Digital (209, 258, 1553)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 27, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (182, 231, 1539)-net over F4, using
- trace code for nets [i] based on digital (28, 77, 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, 77, 513)-net over F64, using
- digital (3, 27, 14)-net over F4, using
(209, 258, 11960)-Net over F4 — Digital
Digital (209, 258, 11960)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4258, 11960, F4, 49) (dual of [11960, 11702, 50]-code), using
- discarding factors / shortening the dual code based on linear OA(4258, 16404, F4, 49) (dual of [16404, 16146, 50]-code), using
- construction X applied to C([0,24]) ⊂ C([0,22]) [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(4239, 16385, F4, 45) (dual of [16385, 16146, 46]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,22], and minimum distance d ≥ |{−22,−21,…,22}|+1 = 46 (BCH-bound) [i]
- linear OA(45, 19, F4, 3) (dual of [19, 14, 4]-code or 19-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([0,22]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4258, 16404, F4, 49) (dual of [16404, 16146, 50]-code), using
(209, 258, large)-Net in Base 4 — Upper bound on s
There is no (209, 258, large)-net in base 4, because
- 47 times m-reduction [i] would yield (209, 211, large)-net in base 4, but