Best Known (206, 206+37, s)-Nets in Base 2
(206, 206+37, 624)-Net over F2 — Constructive and digital
Digital (206, 243, 624)-net over F2, using
- 23 times duplication [i] based on digital (203, 240, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 40, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 40, 104)-net over F64, using
(206, 206+37, 2056)-Net over F2 — Digital
Digital (206, 243, 2056)-net over F2, using
- 22 times duplication [i] based on digital (204, 241, 2056)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2241, 2056, F2, 4, 37) (dual of [(2056, 4), 7983, 38]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2241, 8224, F2, 37) (dual of [8224, 7983, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2241, 8225, F2, 37) (dual of [8225, 7984, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(2235, 8193, F2, 37) (dual of [8193, 7958, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(2209, 8193, F2, 33) (dual of [8193, 7984, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2241, 8225, F2, 37) (dual of [8225, 7984, 38]-code), using
- OOA 4-folding [i] based on linear OA(2241, 8224, F2, 37) (dual of [8224, 7983, 38]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2241, 2056, F2, 4, 37) (dual of [(2056, 4), 7983, 38]-NRT-code), using
(206, 206+37, 84173)-Net in Base 2 — Upper bound on s
There is no (206, 243, 84174)-net in base 2, because
- 1 times m-reduction [i] would yield (206, 242, 84174)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 7 068425 670456 739194 524115 688251 688737 377138 205477 115540 352174 314254 508328 > 2242 [i]