Best Known (209−27, 209, s)-Nets in Base 2
(209−27, 209, 5041)-Net over F2 — Constructive and digital
Digital (182, 209, 5041)-net over F2, using
- net defined by OOA [i] based on linear OOA(2209, 5041, F2, 27, 27) (dual of [(5041, 27), 135898, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2209, 65534, F2, 27) (dual of [65534, 65325, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2209, 65534, F2, 27) (dual of [65534, 65325, 28]-code), using
(209−27, 209, 9406)-Net over F2 — Digital
Digital (182, 209, 9406)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2209, 9406, F2, 6, 27) (dual of [(9406, 6), 56227, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2209, 10922, F2, 6, 27) (dual of [(10922, 6), 65323, 28]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2209, 65532, F2, 27) (dual of [65532, 65323, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- discarding factors / shortening the dual code based on linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using
- OOA 6-folding [i] based on linear OA(2209, 65532, F2, 27) (dual of [65532, 65323, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(2209, 10922, F2, 6, 27) (dual of [(10922, 6), 65323, 28]-NRT-code), using
(209−27, 209, 371418)-Net in Base 2 — Upper bound on s
There is no (182, 209, 371419)-net in base 2, because
- 1 times m-reduction [i] would yield (182, 208, 371419)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 411 378884 055179 977681 503580 313495 642916 971446 392119 928624 232912 > 2208 [i]