Best Known (49−13, 49, s)-Nets in Base 2
(49−13, 49, 72)-Net over F2 — Constructive and digital
Digital (36, 49, 72)-net over F2, using
- 21 times duplication [i] based on digital (35, 48, 72)-net over F2, using
- trace code for nets [i] based on digital (3, 16, 24)-net over F8, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- the Klein quartic over F8 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 3 and N(F) ≥ 24, using
- net from sequence [i] based on digital (3, 23)-sequence over F8, using
- trace code for nets [i] based on digital (3, 16, 24)-net over F8, using
(49−13, 49, 105)-Net over F2 — Digital
Digital (36, 49, 105)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(249, 105, F2, 2, 13) (dual of [(105, 2), 161, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(249, 128, F2, 2, 13) (dual of [(128, 2), 207, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(249, 256, F2, 13) (dual of [256, 207, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 2-folding [i] based on linear OA(249, 256, F2, 13) (dual of [256, 207, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(249, 128, F2, 2, 13) (dual of [(128, 2), 207, 14]-NRT-code), using
(49−13, 49, 757)-Net in Base 2 — Upper bound on s
There is no (36, 49, 758)-net in base 2, because
- 1 times m-reduction [i] would yield (36, 48, 758)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 281 579876 093632 > 248 [i]