Best Known (221−41, 221, s)-Nets in Base 2
(221−41, 221, 320)-Net over F2 — Constructive and digital
Digital (180, 221, 320)-net over F2, using
- 21 times duplication [i] based on digital (179, 220, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 44, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 44, 64)-net over F32, using
(221−41, 221, 762)-Net over F2 — Digital
Digital (180, 221, 762)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2221, 762, F2, 2, 41) (dual of [(762, 2), 1303, 42]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2221, 1024, F2, 2, 41) (dual of [(1024, 2), 1827, 42]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2221, 2048, F2, 41) (dual of [2048, 1827, 42]-code), using
- an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- OOA 2-folding [i] based on linear OA(2221, 2048, F2, 41) (dual of [2048, 1827, 42]-code), using
- discarding factors / shortening the dual code based on linear OOA(2221, 1024, F2, 2, 41) (dual of [(1024, 2), 1827, 42]-NRT-code), using
(221−41, 221, 16977)-Net in Base 2 — Upper bound on s
There is no (180, 221, 16978)-net in base 2, because
- 1 times m-reduction [i] would yield (180, 220, 16978)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 685283 922878 549103 518424 510227 090549 873675 803691 008937 461998 384096 > 2220 [i]