Best Known (111−19, 111, s)-Nets in Base 2
(111−19, 111, 456)-Net over F2 — Constructive and digital
Digital (92, 111, 456)-net over F2, using
- 21 times duplication [i] based on digital (91, 110, 456)-net over F2, using
- net defined by OOA [i] based on linear OOA(2110, 456, F2, 19, 19) (dual of [(456, 19), 8554, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2110, 4105, F2, 19) (dual of [4105, 3995, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2110, 4109, F2, 19) (dual of [4109, 3999, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(297, 4096, F2, 17) (dual of [4096, 3999, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(2110, 4109, F2, 19) (dual of [4109, 3999, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2110, 4105, F2, 19) (dual of [4105, 3995, 20]-code), using
- net defined by OOA [i] based on linear OOA(2110, 456, F2, 19, 19) (dual of [(456, 19), 8554, 20]-NRT-code), using
(111−19, 111, 1027)-Net over F2 — Digital
Digital (92, 111, 1027)-net over F2, using
- 21 times duplication [i] based on digital (91, 110, 1027)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2110, 1027, F2, 4, 19) (dual of [(1027, 4), 3998, 20]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2110, 4108, F2, 19) (dual of [4108, 3998, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2110, 4109, F2, 19) (dual of [4109, 3999, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2109, 4096, F2, 19) (dual of [4096, 3987, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(297, 4096, F2, 17) (dual of [4096, 3999, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(2110, 4109, F2, 19) (dual of [4109, 3999, 20]-code), using
- OOA 4-folding [i] based on linear OA(2110, 4108, F2, 19) (dual of [4108, 3998, 20]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2110, 1027, F2, 4, 19) (dual of [(1027, 4), 3998, 20]-NRT-code), using
(111−19, 111, 19802)-Net in Base 2 — Upper bound on s
There is no (92, 111, 19803)-net in base 2, because
- 1 times m-reduction [i] would yield (92, 110, 19803)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1298 315254 091689 113721 367510 368668 > 2110 [i]