Best Known (135−19, 135, s)-Nets in Base 2
(135−19, 135, 1824)-Net over F2 — Constructive and digital
Digital (116, 135, 1824)-net over F2, using
- 21 times duplication [i] based on digital (115, 134, 1824)-net over F2, using
- net defined by OOA [i] based on linear OOA(2134, 1824, F2, 19, 19) (dual of [(1824, 19), 34522, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2134, 16417, F2, 19) (dual of [16417, 16283, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2134, 16419, F2, 19) (dual of [16419, 16285, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(299, 16384, F2, 15) (dual of [16384, 16285, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-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 Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2134, 16419, F2, 19) (dual of [16419, 16285, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2134, 16417, F2, 19) (dual of [16417, 16283, 20]-code), using
- net defined by OOA [i] based on linear OOA(2134, 1824, F2, 19, 19) (dual of [(1824, 19), 34522, 20]-NRT-code), using
(135−19, 135, 3945)-Net over F2 — Digital
Digital (116, 135, 3945)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2135, 3945, F2, 4, 19) (dual of [(3945, 4), 15645, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2135, 4105, F2, 4, 19) (dual of [(4105, 4), 16285, 20]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2135, 16420, F2, 19) (dual of [16420, 16285, 20]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2134, 16419, F2, 19) (dual of [16419, 16285, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(2127, 16384, F2, 19) (dual of [16384, 16257, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(299, 16384, F2, 15) (dual of [16384, 16285, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(27, 35, F2, 3) (dual of [35, 28, 4]-code or 35-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 Ce(18) ⊂ Ce(14) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2134, 16419, F2, 19) (dual of [16419, 16285, 20]-code), using
- OOA 4-folding [i] based on linear OA(2135, 16420, F2, 19) (dual of [16420, 16285, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(2135, 4105, F2, 4, 19) (dual of [(4105, 4), 16285, 20]-NRT-code), using
(135−19, 135, 125808)-Net in Base 2 — Upper bound on s
There is no (116, 135, 125809)-net in base 2, because
- 1 times m-reduction [i] would yield (116, 134, 125809)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 21779 414807 944633 827958 436427 139832 374116 > 2134 [i]