Best Known (109, 109+21, s)-Nets in Base 2
(109, 109+21, 412)-Net over F2 — Constructive and digital
Digital (109, 130, 412)-net over F2, using
- 23 times duplication [i] based on digital (106, 127, 412)-net over F2, using
- net defined by OOA [i] based on linear OOA(2127, 412, F2, 21, 21) (dual of [(412, 21), 8525, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2127, 4121, F2, 21) (dual of [4121, 3994, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2127, 4127, F2, 21) (dual of [4127, 4000, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2121, 4097, F2, 21) (dual of [4097, 3976, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(297, 4097, F2, 17) (dual of [4097, 4000, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on 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(2127, 4127, F2, 21) (dual of [4127, 4000, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2127, 4121, F2, 21) (dual of [4121, 3994, 22]-code), using
- net defined by OOA [i] based on linear OOA(2127, 412, F2, 21, 21) (dual of [(412, 21), 8525, 22]-NRT-code), using
(109, 109+21, 1249)-Net over F2 — Digital
Digital (109, 130, 1249)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2130, 1249, F2, 3, 21) (dual of [(1249, 3), 3617, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2130, 1376, F2, 3, 21) (dual of [(1376, 3), 3998, 22]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2128, 1376, F2, 3, 21) (dual of [(1376, 3), 4000, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2128, 4128, F2, 21) (dual of [4128, 4000, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2127, 4127, F2, 21) (dual of [4127, 4000, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2121, 4097, F2, 21) (dual of [4097, 3976, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(297, 4097, F2, 17) (dual of [4097, 4000, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on 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 code embedding in larger space [i] based on linear OA(2127, 4127, F2, 21) (dual of [4127, 4000, 22]-code), using
- OOA 3-folding [i] based on linear OA(2128, 4128, F2, 21) (dual of [4128, 4000, 22]-code), using
- 22 times duplication [i] based on linear OOA(2128, 1376, F2, 3, 21) (dual of [(1376, 3), 4000, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2130, 1376, F2, 3, 21) (dual of [(1376, 3), 3998, 22]-NRT-code), using
(109, 109+21, 34600)-Net in Base 2 — Upper bound on s
There is no (109, 130, 34601)-net in base 2, because
- 1 times m-reduction [i] would yield (109, 129, 34601)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 680 679572 127867 559872 210067 479970 474644 > 2129 [i]