Best Known (189, 189+43, s)-Nets in Base 2
(189, 189+43, 320)-Net over F2 — Constructive and digital
Digital (189, 232, 320)-net over F2, using
- 22 times duplication [i] based on digital (187, 230, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 46, 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, 46, 64)-net over F32, using
(189, 189+43, 792)-Net over F2 — Digital
Digital (189, 232, 792)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2232, 792, F2, 2, 43) (dual of [(792, 2), 1352, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2232, 1024, F2, 2, 43) (dual of [(1024, 2), 1816, 44]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2232, 2048, F2, 43) (dual of [2048, 1816, 44]-code), using
- an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- OOA 2-folding [i] based on linear OA(2232, 2048, F2, 43) (dual of [2048, 1816, 44]-code), using
- discarding factors / shortening the dual code based on linear OOA(2232, 1024, F2, 2, 43) (dual of [(1024, 2), 1816, 44]-NRT-code), using
(189, 189+43, 17744)-Net in Base 2 — Upper bound on s
There is no (189, 232, 17745)-net in base 2, because
- 1 times m-reduction [i] would yield (189, 231, 17745)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 3452 695854 840284 739350 210889 116578 517495 670326 513377 237769 096898 833472 > 2231 [i]