Best Known (137−21, 137, s)-Nets in Base 2
(137−21, 137, 822)-Net over F2 — Constructive and digital
Digital (116, 137, 822)-net over F2, using
- net defined by OOA [i] based on linear OOA(2137, 822, F2, 21, 21) (dual of [(822, 21), 17125, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2137, 8221, F2, 21) (dual of [8221, 8084, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2137, 8225, F2, 21) (dual of [8225, 8088, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2131, 8193, F2, 21) (dual of [8193, 8062, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2105, 8193, F2, 17) (dual of [8193, 8088, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2137, 8225, F2, 21) (dual of [8225, 8088, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2137, 8221, F2, 21) (dual of [8221, 8084, 22]-code), using
(137−21, 137, 2056)-Net over F2 — Digital
Digital (116, 137, 2056)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2137, 2056, F2, 4, 21) (dual of [(2056, 4), 8087, 22]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2137, 8224, F2, 21) (dual of [8224, 8087, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2137, 8225, F2, 21) (dual of [8225, 8088, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2131, 8193, F2, 21) (dual of [8193, 8062, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2105, 8193, F2, 17) (dual of [8193, 8088, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (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([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2137, 8225, F2, 21) (dual of [8225, 8088, 22]-code), using
- OOA 4-folding [i] based on linear OA(2137, 8224, F2, 21) (dual of [8224, 8087, 22]-code), using
(137−21, 137, 56217)-Net in Base 2 — Upper bound on s
There is no (116, 137, 56218)-net in base 2, because
- 1 times m-reduction [i] would yield (116, 136, 56218)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 87118 573805 902611 933246 117013 325402 818864 > 2136 [i]