Best Known (85−23, 85, s)-Nets in Base 2
(85−23, 85, 84)-Net over F2 — Constructive and digital
Digital (62, 85, 84)-net over F2, using
- 21 times duplication [i] based on digital (61, 84, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 28, 28)-net over F8, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 5 and N(F) ≥ 28, using
- net from sequence [i] based on digital (5, 27)-sequence over F8, using
- trace code for nets [i] based on digital (5, 28, 28)-net over F8, using
(85−23, 85, 122)-Net over F2 — Digital
Digital (62, 85, 122)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(285, 122, F2, 2, 23) (dual of [(122, 2), 159, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(285, 128, F2, 2, 23) (dual of [(128, 2), 171, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(285, 256, F2, 23) (dual of [256, 171, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- OOA 2-folding [i] based on linear OA(285, 256, F2, 23) (dual of [256, 171, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(285, 128, F2, 2, 23) (dual of [(128, 2), 171, 24]-NRT-code), using
(85−23, 85, 960)-Net in Base 2 — Upper bound on s
There is no (62, 85, 961)-net in base 2, because
- 1 times m-reduction [i] would yield (62, 84, 961)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 19 369295 623270 141575 955776 > 284 [i]