Best Known (188, 188+21, s)-Nets in Base 2
(188, 188+21, 104862)-Net over F2 — Constructive and digital
Digital (188, 209, 104862)-net over F2, using
- 21 times duplication [i] based on digital (187, 208, 104862)-net over F2, using
- net defined by OOA [i] based on linear OOA(2208, 104862, F2, 21, 21) (dual of [(104862, 21), 2201894, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2208, 1048621, F2, 21) (dual of [1048621, 1048413, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, 1048624, F2, 21) (dual of [1048624, 1048416, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2201, 1048577, F2, 21) (dual of [1048577, 1048376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2161, 1048577, F2, 17) (dual of [1048577, 1048416, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(27, 47, F2, 3) (dual of [47, 40, 4]-code or 47-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2208, 1048624, F2, 21) (dual of [1048624, 1048416, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2208, 1048621, F2, 21) (dual of [1048621, 1048413, 22]-code), using
- net defined by OOA [i] based on linear OOA(2208, 104862, F2, 21, 21) (dual of [(104862, 21), 2201894, 22]-NRT-code), using
(188, 188+21, 149803)-Net over F2 — Digital
Digital (188, 209, 149803)-net over F2, using
- 21 times duplication [i] based on digital (187, 208, 149803)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2208, 149803, F2, 7, 21) (dual of [(149803, 7), 1048413, 22]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2208, 1048621, F2, 21) (dual of [1048621, 1048413, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, 1048624, F2, 21) (dual of [1048624, 1048416, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2201, 1048577, F2, 21) (dual of [1048577, 1048376, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2161, 1048577, F2, 17) (dual of [1048577, 1048416, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(27, 47, F2, 3) (dual of [47, 40, 4]-code or 47-cap in PG(6,2)), using
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 4 [i]
- discarding factors / shortening the dual code based on linear OA(27, 63, F2, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,2)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2208, 1048624, F2, 21) (dual of [1048624, 1048416, 22]-code), using
- OOA 7-folding [i] based on linear OA(2208, 1048621, F2, 21) (dual of [1048621, 1048413, 22]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2208, 149803, F2, 7, 21) (dual of [(149803, 7), 1048413, 22]-NRT-code), using
(188, 188+21, 8267980)-Net in Base 2 — Upper bound on s
There is no (188, 209, 8267981)-net in base 2, because
- 1 times m-reduction [i] would yield (188, 208, 8267981)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 411 376314 415328 035758 448570 626128 365865 899312 310490 030483 480078 > 2208 [i]