Best Known (24, 24+9, s)-Nets in Base 2
(24, 24+9, 64)-Net over F2 — Constructive and digital
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
(24, 24+9, 100)-Net over F2 — Digital
Digital (24, 33, 100)-net over F2, using
- net defined by OOA [i] based on linear OOA(233, 100, F2, 9, 9) (dual of [(100, 9), 867, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(233, 100, F2, 8, 9) (dual of [(100, 8), 767, 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
- appending kth column [i] based on linear OOA(233, 100, F2, 8, 9) (dual of [(100, 8), 767, 10]-NRT-code), using
(24, 24+9, 561)-Net in Base 2 — Upper bound on s
There is no (24, 33, 562)-net in base 2, because
- 1 times m-reduction [i] would yield (24, 32, 562)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 4321 000788 > 232 [i]