Best Known (37−9, 37, s)-Nets in Base 2
(37−9, 37, 128)-Net over F2 — Constructive and digital
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
(37−9, 37, 171)-Net over F2 — Digital
Digital (28, 37, 171)-net over F2, using
- net defined by OOA [i] based on linear OOA(237, 171, F2, 9, 9) (dual of [(171, 9), 1502, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(237, 171, F2, 8, 9) (dual of [(171, 8), 1331, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(237, 171, F2, 3, 9) (dual of [(171, 3), 476, 10]-NRT-code), using
- OOA 3-folding [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 3-folding [i] based on linear OA(237, 513, F2, 9) (dual of [513, 476, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(237, 171, F2, 3, 9) (dual of [(171, 3), 476, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(237, 171, F2, 8, 9) (dual of [(171, 8), 1331, 10]-NRT-code), using
(37−9, 37, 1127)-Net in Base 2 — Upper bound on s
There is no (28, 37, 1128)-net in base 2, because
- 1 times m-reduction [i] would yield (28, 36, 1128)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 68779 309979 > 236 [i]