Best Known (221, 221+35, s)-Nets in Base 2
(221, 221+35, 1927)-Net over F2 — Constructive and digital
Digital (221, 256, 1927)-net over F2, using
- net defined by OOA [i] based on linear OOA(2256, 1927, F2, 35, 35) (dual of [(1927, 35), 67189, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2256, 32760, F2, 35) (dual of [32760, 32504, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2256, 32768, F2, 35) (dual of [32768, 32512, 36]-code), using
- an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- discarding factors / shortening the dual code based on linear OA(2256, 32768, F2, 35) (dual of [32768, 32512, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2256, 32760, F2, 35) (dual of [32760, 32504, 36]-code), using
(221, 221+35, 5461)-Net over F2 — Digital
Digital (221, 256, 5461)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2256, 5461, F2, 6, 35) (dual of [(5461, 6), 32510, 36]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2256, 32766, F2, 35) (dual of [32766, 32510, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2256, 32768, F2, 35) (dual of [32768, 32512, 36]-code), using
- an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- discarding factors / shortening the dual code based on linear OA(2256, 32768, F2, 35) (dual of [32768, 32512, 36]-code), using
- OOA 6-folding [i] based on linear OA(2256, 32766, F2, 35) (dual of [32766, 32510, 36]-code), using
(221, 221+35, 235152)-Net in Base 2 — Upper bound on s
There is no (221, 256, 235153)-net in base 2, because
- 1 times m-reduction [i] would yield (221, 255, 235153)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 57899 554642 374068 286339 828069 730346 803582 517639 198983 318693 638217 183188 719634 > 2255 [i]