Best Known (210, 210+18, s)-Nets in Base 2
(210, 210+18, 932096)-Net over F2 — Constructive and digital
Digital (210, 228, 932096)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (12, 21, 29)-net over F2, using
- digital (189, 207, 932067)-net over F2, using
- net defined by OOA [i] based on linear OOA(2207, 932067, F2, 18, 18) (dual of [(932067, 18), 16776999, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- net defined by OOA [i] based on linear OOA(2207, 932067, F2, 18, 18) (dual of [(932067, 18), 16776999, 19]-NRT-code), using
(210, 210+18, 1677749)-Net over F2 — Digital
Digital (210, 228, 1677749)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2228, 1677749, F2, 5, 18) (dual of [(1677749, 5), 8388517, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(221, 29, F2, 5, 9) (dual of [(29, 5), 124, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (12, 21, 29)-net over F2, using
- linear OOA(2207, 1677720, F2, 5, 18) (dual of [(1677720, 5), 8388393, 19]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- the primitive narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- discarding factors / shortening the dual code based on linear OA(2207, large, F2, 18) (dual of [large, large−207, 19]-code), using
- OOA 5-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- linear OOA(221, 29, F2, 5, 9) (dual of [(29, 5), 124, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(210, 210+18, large)-Net in Base 2 — Upper bound on s
There is no (210, 228, large)-net in base 2, because
- 16 times m-reduction [i] would yield (210, 212, large)-net in base 2, but