Best Known (225−33, 225, s)-Nets in Base 2
(225−33, 225, 1024)-Net over F2 — Constructive and digital
Digital (192, 225, 1024)-net over F2, using
- net defined by OOA [i] based on linear OOA(2225, 1024, F2, 33, 33) (dual of [(1024, 33), 33567, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2225, 16385, F2, 33) (dual of [16385, 16160, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(2225, 16385, F2, 33) (dual of [16385, 16160, 34]-code), using
(225−33, 225, 3060)-Net over F2 — Digital
Digital (192, 225, 3060)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2225, 3060, F2, 5, 33) (dual of [(3060, 5), 15075, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2225, 3277, F2, 5, 33) (dual of [(3277, 5), 16160, 34]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2225, 16385, F2, 33) (dual of [16385, 16160, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 5-folding [i] based on linear OA(2225, 16385, F2, 33) (dual of [16385, 16160, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(2225, 3277, F2, 5, 33) (dual of [(3277, 5), 16160, 34]-NRT-code), using
(225−33, 225, 111395)-Net in Base 2 — Upper bound on s
There is no (192, 225, 111396)-net in base 2, because
- 1 times m-reduction [i] would yield (192, 224, 111396)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 26 962462 205674 449233 262378 114514 382168 209193 342736 075899 579677 228102 > 2224 [i]