Best Known (91−21, 91, s)-Nets in Base 2
(91−21, 91, 135)-Net over F2 — Constructive and digital
Digital (70, 91, 135)-net over F2, using
- 21 times duplication [i] based on digital (69, 90, 135)-net over F2, using
- trace code for nets [i] based on digital (9, 30, 45)-net over F8, using
- net from sequence [i] based on digital (9, 44)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 9 and N(F) ≥ 45, using
- net from sequence [i] based on digital (9, 44)-sequence over F8, using
- trace code for nets [i] based on digital (9, 30, 45)-net over F8, using
(91−21, 91, 208)-Net over F2 — Digital
Digital (70, 91, 208)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(291, 208, F2, 2, 21) (dual of [(208, 2), 325, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(291, 256, F2, 2, 21) (dual of [(256, 2), 421, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(291, 512, F2, 21) (dual of [512, 421, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- OOA 2-folding [i] based on linear OA(291, 512, F2, 21) (dual of [512, 421, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(291, 256, F2, 2, 21) (dual of [(256, 2), 421, 22]-NRT-code), using
(91−21, 91, 2304)-Net in Base 2 — Upper bound on s
There is no (70, 91, 2305)-net in base 2, because
- 1 times m-reduction [i] would yield (70, 90, 2305)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1241 952547 721257 670295 761664 > 290 [i]