Best Known (91−18, 91, s)-Nets in Base 2
(91−18, 91, 180)-Net over F2 — Constructive and digital
Digital (73, 91, 180)-net over F2, using
- 1 times m-reduction [i] based on digital (73, 92, 180)-net over F2, using
- trace code for nets [i] based on digital (4, 23, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- trace code for nets [i] based on digital (4, 23, 45)-net over F16, using
(91−18, 91, 372)-Net over F2 — Digital
Digital (73, 91, 372)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(291, 372, F2, 2, 18) (dual of [(372, 2), 653, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(291, 517, F2, 2, 18) (dual of [(517, 2), 943, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(291, 1034, F2, 18) (dual of [1034, 943, 19]-code), using
- 1 times truncation [i] based on linear OA(292, 1035, F2, 19) (dual of [1035, 943, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(291, 1024, F2, 19) (dual of [1024, 933, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(281, 1024, F2, 17) (dual of [1024, 943, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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(18) ⊂ Ce(16) [i] based on
- 1 times truncation [i] based on linear OA(292, 1035, F2, 19) (dual of [1035, 943, 20]-code), using
- OOA 2-folding [i] based on linear OA(291, 1034, F2, 18) (dual of [1034, 943, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(291, 517, F2, 2, 18) (dual of [(517, 2), 943, 19]-NRT-code), using
(91−18, 91, 4573)-Net in Base 2 — Upper bound on s
There is no (73, 91, 4574)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 2477 317406 271932 570219 129847 > 291 [i]