Best Known (226, 226+18, s)-Nets in Base 2
(226, 226+18, 932195)-Net over F2 — Constructive and digital
Digital (226, 244, 932195)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (28, 37, 128)-net over F2, using
- net defined by OOA [i] based on linear OOA(237, 128, F2, 9, 9) (dual of [(128, 9), 1115, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(237, 128, F2, 8, 9) (dual of [(128, 8), 987, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(237, 513, F2, 9) (dual of [513, 476, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 513 | 218−1, 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(237, 513, F2, 9) (dual of [513, 476, 10]-code), using
- appending kth column [i] based on linear OOA(237, 128, F2, 8, 9) (dual of [(128, 8), 987, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(237, 128, F2, 9, 9) (dual of [(128, 9), 1115, 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 (28, 37, 128)-net over F2, using
(226, 226+18, 2045875)-Net over F2 — Digital
Digital (226, 244, 2045875)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2244, 2045875, F2, 4, 18) (dual of [(2045875, 4), 8183256, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2244, 2097278, F2, 4, 18) (dual of [(2097278, 4), 8388868, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(2244, 4194556, F2, 2, 18) (dual of [(4194556, 2), 8388868, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2244, 4194557, F2, 2, 18) (dual of [(4194557, 2), 8388870, 19]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(237, 256, F2, 2, 9) (dual of [(256, 2), 475, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(237, 512, F2, 9) (dual of [512, 475, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 2-folding [i] based on linear OA(237, 512, F2, 9) (dual of [512, 475, 10]-code), using
- linear OOA(2207, 4194301, F2, 2, 18) (dual of [(4194301, 2), 8388395, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2207, 8388602, F2, 18) (dual of [8388602, 8388395, 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 2-folding [i] based on linear OA(2207, 8388602, F2, 18) (dual of [8388602, 8388395, 19]-code), using
- linear OOA(237, 256, F2, 2, 9) (dual of [(256, 2), 475, 10]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(2244, 4194557, F2, 2, 18) (dual of [(4194557, 2), 8388870, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OOA(2244, 4194556, F2, 2, 18) (dual of [(4194556, 2), 8388868, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2244, 2097278, F2, 4, 18) (dual of [(2097278, 4), 8388868, 19]-NRT-code), using
(226, 226+18, large)-Net in Base 2 — Upper bound on s
There is no (226, 244, large)-net in base 2, because
- 16 times m-reduction [i] would yield (226, 228, large)-net in base 2, but