Best Known (136−20, 136, s)-Nets in Base 2
(136−20, 136, 822)-Net over F2 — Constructive and digital
Digital (116, 136, 822)-net over F2, using
- net defined by OOA [i] based on linear OOA(2136, 822, F2, 20, 20) (dual of [(822, 20), 16304, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(2136, 8220, F2, 20) (dual of [8220, 8084, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2136, 8224, F2, 20) (dual of [8224, 8088, 21]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2137, 8225, F2, 21) (dual of [8225, 8088, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2136, 8224, F2, 20) (dual of [8224, 8088, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(2136, 8220, F2, 20) (dual of [8220, 8084, 21]-code), using
(136−20, 136, 2139)-Net over F2 — Digital
Digital (116, 136, 2139)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2136, 2139, F2, 3, 20) (dual of [(2139, 3), 6281, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2136, 2741, F2, 3, 20) (dual of [(2741, 3), 8087, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2136, 8223, F2, 20) (dual of [8223, 8087, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2136, 8224, F2, 20) (dual of [8224, 8088, 21]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2137, 8225, F2, 21) (dual of [8225, 8088, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2136, 8224, F2, 20) (dual of [8224, 8088, 21]-code), using
- OOA 3-folding [i] based on linear OA(2136, 8223, F2, 20) (dual of [8223, 8087, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(2136, 2741, F2, 3, 20) (dual of [(2741, 3), 8087, 21]-NRT-code), using
(136−20, 136, 56217)-Net in Base 2 — Upper bound on s
There is no (116, 136, 56218)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 87118 573805 902611 933246 117013 325402 818864 > 2136 [i]