Best Known (234, 252, s)-Nets in Base 2
(234, 252, 932579)-Net over F2 — Constructive and digital
Digital (234, 252, 932579)-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, 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
- digital (36, 45, 512)-net over F2, using
(234, 252, 2097833)-Net over F2 — Digital
Digital (234, 252, 2097833)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2252, 2097833, F2, 4, 18) (dual of [(2097833, 4), 8391080, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(245, 683, F2, 4, 9) (dual of [(683, 4), 2687, 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(2207, 2097150, F2, 4, 18) (dual of [(2097150, 4), 8388393, 19]-NRT-code), using
- OOA 4-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 4-folding [i] based on linear OA(2207, 8388600, F2, 18) (dual of [8388600, 8388393, 19]-code), using
- linear OOA(245, 683, F2, 4, 9) (dual of [(683, 4), 2687, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(234, 252, large)-Net in Base 2 — Upper bound on s
There is no (234, 252, large)-net in base 2, because
- 16 times m-reduction [i] would yield (234, 236, large)-net in base 2, but