Best Known (222, 222+18, s)-Nets in Base 2
(222, 222+18, 932131)-Net over F2 — Constructive and digital
Digital (222, 240, 932131)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (24, 33, 64)-net over F2, using
- net defined by OOA [i] based on linear OOA(233, 64, F2, 9, 9) (dual of [(64, 9), 543, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(233, 64, F2, 8, 9) (dual of [(64, 8), 479, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(233, 257, F2, 9) (dual of [257, 224, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 257 | 216−1, 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(233, 257, F2, 9) (dual of [257, 224, 10]-code), using
- appending kth column [i] based on linear OOA(233, 64, F2, 8, 9) (dual of [(64, 8), 479, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(233, 64, F2, 9, 9) (dual of [(64, 9), 543, 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 (24, 33, 64)-net over F2, using
(222, 222+18, 1677820)-Net over F2 — Digital
Digital (222, 240, 1677820)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2240, 1677820, F2, 5, 18) (dual of [(1677820, 5), 8388860, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(233, 100, F2, 5, 9) (dual of [(100, 5), 467, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(233, 100, F2, 2, 9) (dual of [(100, 2), 167, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(233, 128, F2, 2, 9) (dual of [(128, 2), 223, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(233, 256, F2, 9) (dual of [256, 223, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 2-folding [i] based on linear OA(233, 256, F2, 9) (dual of [256, 223, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(233, 128, F2, 2, 9) (dual of [(128, 2), 223, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(233, 100, F2, 2, 9) (dual of [(100, 2), 167, 10]-NRT-code), 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(233, 100, F2, 5, 9) (dual of [(100, 5), 467, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
(222, 222+18, large)-Net in Base 2 — Upper bound on s
There is no (222, 240, large)-net in base 2, because
- 16 times m-reduction [i] would yield (222, 224, large)-net in base 2, but