Best Known (220−23, 220, s)-Nets in Base 2
(220−23, 220, 47666)-Net over F2 — Constructive and digital
Digital (197, 220, 47666)-net over F2, using
- 23 times duplication [i] based on digital (194, 217, 47666)-net over F2, using
- net defined by OOA [i] based on linear OOA(2217, 47666, F2, 23, 23) (dual of [(47666, 23), 1096101, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2217, 524327, F2, 23) (dual of [524327, 524110, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2217, 524333, F2, 23) (dual of [524333, 524116, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2210, 524288, F2, 23) (dual of [524288, 524078, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2172, 524288, F2, 19) (dual of [524288, 524116, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-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(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2217, 524333, F2, 23) (dual of [524333, 524116, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2217, 524327, F2, 23) (dual of [524327, 524110, 24]-code), using
- net defined by OOA [i] based on linear OOA(2217, 47666, F2, 23, 23) (dual of [(47666, 23), 1096101, 24]-NRT-code), using
(220−23, 220, 74905)-Net over F2 — Digital
Digital (197, 220, 74905)-net over F2, using
- 21 times duplication [i] based on digital (196, 219, 74905)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2219, 74905, F2, 7, 23) (dual of [(74905, 7), 524116, 24]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2219, 524335, F2, 23) (dual of [524335, 524116, 24]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2217, 524333, F2, 23) (dual of [524333, 524116, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(2210, 524288, F2, 23) (dual of [524288, 524078, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2172, 524288, F2, 19) (dual of [524288, 524116, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(27, 45, F2, 3) (dual of [45, 38, 4]-code or 45-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(22) ⊂ Ce(18) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2217, 524333, F2, 23) (dual of [524333, 524116, 24]-code), using
- OOA 7-folding [i] based on linear OA(2219, 524335, F2, 23) (dual of [524335, 524116, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2219, 74905, F2, 7, 23) (dual of [(74905, 7), 524116, 24]-NRT-code), using
(220−23, 220, 4833328)-Net in Base 2 — Upper bound on s
There is no (197, 220, 4833329)-net in base 2, because
- 1 times m-reduction [i] would yield (197, 219, 4833329)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 842500 245051 512475 782000 709973 567758 306226 319115 369635 310456 510160 > 2219 [i]