Best Known (224, 257, s)-Nets in Base 2
(224, 257, 4096)-Net over F2 — Constructive and digital
Digital (224, 257, 4096)-net over F2, using
- net defined by OOA [i] based on linear OOA(2257, 4096, F2, 33, 33) (dual of [(4096, 33), 134911, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(2257, 65537, F2, 33) (dual of [65537, 65280, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 232−1, 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(2257, 65537, F2, 33) (dual of [65537, 65280, 34]-code), using
(224, 257, 9362)-Net over F2 — Digital
Digital (224, 257, 9362)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2257, 9362, F2, 7, 33) (dual of [(9362, 7), 65277, 34]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2257, 65534, F2, 33) (dual of [65534, 65277, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2257, 65536, F2, 33) (dual of [65536, 65279, 34]-code), using
- an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- discarding factors / shortening the dual code based on linear OA(2257, 65536, F2, 33) (dual of [65536, 65279, 34]-code), using
- OOA 7-folding [i] based on linear OA(2257, 65534, F2, 33) (dual of [65534, 65277, 34]-code), using
(224, 257, 445651)-Net in Base 2 — Upper bound on s
There is no (224, 257, 445652)-net in base 2, because
- 1 times m-reduction [i] would yield (224, 256, 445652)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 115792 535477 305665 208780 595996 770576 530233 275303 579493 464366 450336 629749 046189 > 2256 [i]