Best Known (49−9, 49, s)-Nets in Base 2
(49−9, 49, 1024)-Net over F2 — Constructive and digital
Digital (40, 49, 1024)-net over F2, using
- net defined by OOA [i] based on linear OOA(249, 1024, F2, 9, 9) (dual of [(1024, 9), 9167, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(249, 1024, F2, 8, 9) (dual of [(1024, 8), 8143, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(249, 4097, F2, 9) (dual of [4097, 4048, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, 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(249, 4097, F2, 9) (dual of [4097, 4048, 10]-code), using
- appending kth column [i] based on linear OOA(249, 1024, F2, 8, 9) (dual of [(1024, 8), 8143, 10]-NRT-code), using
(49−9, 49, 1365)-Net over F2 — Digital
Digital (40, 49, 1365)-net over F2, using
- net defined by OOA [i] based on linear OOA(249, 1365, F2, 9, 9) (dual of [(1365, 9), 12236, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(249, 1365, F2, 8, 9) (dual of [(1365, 8), 10871, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(249, 1365, F2, 3, 9) (dual of [(1365, 3), 4046, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(249, 4095, F2, 9) (dual of [4095, 4046, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using
- OOA 3-folding [i] based on linear OA(249, 4095, F2, 9) (dual of [4095, 4046, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(249, 1365, F2, 3, 9) (dual of [(1365, 3), 4046, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(249, 1365, F2, 8, 9) (dual of [(1365, 8), 10871, 10]-NRT-code), using
(49−9, 49, 9060)-Net in Base 2 — Upper bound on s
There is no (40, 49, 9061)-net in base 2, because
- 1 times m-reduction [i] would yield (40, 48, 9061)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 281 544569 567561 > 248 [i]