Best Known (216−33, 216, s)-Nets in Base 2
(216−33, 216, 624)-Net over F2 — Constructive and digital
Digital (183, 216, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 36, 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
(216−33, 216, 2056)-Net over F2 — Digital
Digital (183, 216, 2056)-net over F2, using
- 21 times duplication [i] based on digital (182, 215, 2056)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2215, 2056, F2, 4, 33) (dual of [(2056, 4), 8009, 34]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2215, 8224, F2, 33) (dual of [8224, 8009, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2215, 8225, F2, 33) (dual of [8225, 8010, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- 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(2183, 8193, F2, 29) (dual of [8193, 8010, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (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([0,16]) ⊂ C([0,14]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2215, 8225, F2, 33) (dual of [8225, 8010, 34]-code), using
- OOA 4-folding [i] based on linear OA(2215, 8224, F2, 33) (dual of [8224, 8009, 34]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2215, 2056, F2, 4, 33) (dual of [(2056, 4), 8009, 34]-NRT-code), using
(216−33, 216, 75421)-Net in Base 2 — Upper bound on s
There is no (183, 216, 75422)-net in base 2, because
- 1 times m-reduction [i] would yield (183, 215, 75422)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 52663 965509 450136 450917 995392 592467 252433 260187 415213 532291 751923 > 2215 [i]