Best Known (253−19, 253, s)-Nets in Base 2
(253−19, 253, 932578)-Net over F2 — Constructive and digital
Digital (234, 253, 932578)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (36, 45, 512)-net over F2, using
- net defined by OOA [i] based on linear OOA(245, 512, F2, 9, 9) (dual of [(512, 9), 4563, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(245, 512, F2, 8, 9) (dual of [(512, 8), 4051, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- appending kth column [i] based on linear OOA(245, 512, F2, 8, 9) (dual of [(512, 8), 4051, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(245, 512, F2, 9, 9) (dual of [(512, 9), 4563, 10]-NRT-code), 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
- digital (36, 45, 512)-net over F2, using
(253−19, 253, 1678403)-Net over F2 — Digital
Digital (234, 253, 1678403)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2253, 1678403, F2, 5, 19) (dual of [(1678403, 5), 8391762, 20]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(245, 683, F2, 5, 9) (dual of [(683, 5), 3370, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(245, 683, F2, 3, 9) (dual of [(683, 3), 2004, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(245, 683, F2, 3, 9) (dual of [(683, 3), 2004, 10]-NRT-code), using
- linear OOA(2208, 1677720, F2, 5, 19) (dual of [(1677720, 5), 8388392, 20]-NRT-code), using
- OOA 5-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 5-folding [i] based on linear OA(2208, 8388600, F2, 19) (dual of [8388600, 8388392, 20]-code), using
- linear OOA(245, 683, F2, 5, 9) (dual of [(683, 5), 3370, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(253−19, 253, large)-Net in Base 2 — Upper bound on s
There is no (234, 253, large)-net in base 2, because
- 17 times m-reduction [i] would yield (234, 236, large)-net in base 2, but