Best Known (256−32, 256, s)-Nets in Base 2
(256−32, 256, 4096)-Net over F2 — Constructive and digital
Digital (224, 256, 4096)-net over F2, using
- net defined by OOA [i] based on linear OOA(2256, 4096, F2, 32, 32) (dual of [(4096, 32), 130816, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(2256, 65536, F2, 32) (dual of [65536, 65280, 33]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(2257, 65537, F2, 33) (dual of [65537, 65280, 34]-code), using
- OA 16-folding and stacking [i] based on linear OA(2256, 65536, F2, 32) (dual of [65536, 65280, 33]-code), using
(256−32, 256, 10385)-Net over F2 — Digital
Digital (224, 256, 10385)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2256, 10385, F2, 6, 32) (dual of [(10385, 6), 62054, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2256, 10922, F2, 6, 32) (dual of [(10922, 6), 65276, 33]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2256, 65532, F2, 32) (dual of [65532, 65276, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(2256, 65536, F2, 32) (dual of [65536, 65280, 33]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(2257, 65537, F2, 33) (dual of [65537, 65280, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2256, 65536, F2, 32) (dual of [65536, 65280, 33]-code), using
- OOA 6-folding [i] based on linear OA(2256, 65532, F2, 32) (dual of [65532, 65276, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(2256, 10922, F2, 6, 32) (dual of [(10922, 6), 65276, 33]-NRT-code), using
(256−32, 256, 445651)-Net in Base 2 — Upper bound on s
There is no (224, 256, 445652)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 115792 535477 305665 208780 595996 770576 530233 275303 579493 464366 450336 629749 046189 > 2256 [i]