Best Known (228−19, 228, s)-Nets in Base 2
(228−19, 228, 932092)-Net over F2 — Constructive and digital
Digital (209, 228, 932092)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (11, 20, 26)-net over F2, using
- digital (189, 208, 932066)-net over F2, using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2208, 8388595, F2, 19) (dual of [8388595, 8388387, 20]-code), using
- net defined by OOA [i] based on linear OOA(2208, 932066, F2, 19, 19) (dual of [(932066, 19), 17709046, 20]-NRT-code), using
(228−19, 228, 1398126)-Net over F2 — Digital
Digital (209, 228, 1398126)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2228, 1398126, F2, 6, 19) (dual of [(1398126, 6), 8388528, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(220, 26, F2, 6, 9) (dual of [(26, 6), 136, 10]-NRT-code), using
- extracting embedded OOA [i] based on digital (11, 20, 26)-net over F2, using
- linear OOA(2208, 1398100, F2, 6, 19) (dual of [(1398100, 6), 8388392, 20]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 8388607 = 223−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2208, large, F2, 19) (dual of [large, large−208, 20]-code), using
- OOA 6-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- linear OOA(220, 26, F2, 6, 9) (dual of [(26, 6), 136, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(228−19, 228, large)-Net in Base 2 — Upper bound on s
There is no (209, 228, large)-net in base 2, because
- 17 times m-reduction [i] would yield (209, 211, large)-net in base 2, but