Best Known (66−17, 66, s)-Nets in Base 2
(66−17, 66, 84)-Net over F2 — Constructive and digital
Digital (49, 66, 84)-net over F2, using
- trace code for nets [i] based on digital (5, 22, 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
(66−17, 66, 125)-Net over F2 — Digital
Digital (49, 66, 125)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(266, 125, F2, 2, 17) (dual of [(125, 2), 184, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(266, 132, F2, 2, 17) (dual of [(132, 2), 198, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(266, 264, F2, 17) (dual of [264, 198, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(266, 265, F2, 17) (dual of [265, 199, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- linear OA(265, 256, F2, 17) (dual of [256, 191, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(257, 256, F2, 15) (dual of [256, 199, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(21, 9, F2, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(266, 265, F2, 17) (dual of [265, 199, 18]-code), using
- OOA 2-folding [i] based on linear OA(266, 264, F2, 17) (dual of [264, 198, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(266, 132, F2, 2, 17) (dual of [(132, 2), 198, 18]-NRT-code), using
(66−17, 66, 1039)-Net in Base 2 — Upper bound on s
There is no (49, 66, 1040)-net in base 2, because
- 1 times m-reduction [i] would yield (49, 65, 1040)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 37 048484 975820 270499 > 265 [i]