Best Known (189−29, 189, s)-Nets in Base 2
(189−29, 189, 587)-Net over F2 — Constructive and digital
Digital (160, 189, 587)-net over F2, using
- net defined by OOA [i] based on linear OOA(2189, 587, F2, 29, 29) (dual of [(587, 29), 16834, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2189, 8219, F2, 29) (dual of [8219, 8030, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2189, 8225, F2, 29) (dual of [8225, 8036, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- 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(2157, 8193, F2, 25) (dual of [8193, 8036, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2189, 8225, F2, 29) (dual of [8225, 8036, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2189, 8219, F2, 29) (dual of [8219, 8030, 30]-code), using
(189−29, 189, 2016)-Net over F2 — Digital
Digital (160, 189, 2016)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2189, 2016, F2, 4, 29) (dual of [(2016, 4), 7875, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2189, 2056, F2, 4, 29) (dual of [(2056, 4), 8035, 30]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2189, 8224, F2, 29) (dual of [8224, 8035, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2189, 8225, F2, 29) (dual of [8225, 8036, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- 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(2157, 8193, F2, 25) (dual of [8193, 8036, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (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([0,14]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2189, 8225, F2, 29) (dual of [8225, 8036, 30]-code), using
- OOA 4-folding [i] based on linear OA(2189, 8224, F2, 29) (dual of [8224, 8035, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2189, 2056, F2, 4, 29) (dual of [(2056, 4), 8035, 30]-NRT-code), using
(189−29, 189, 66638)-Net in Base 2 — Upper bound on s
There is no (160, 189, 66639)-net in base 2, because
- 1 times m-reduction [i] would yield (160, 188, 66639)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 392 337764 073104 215140 314167 216575 887960 390356 624079 860261 > 2188 [i]