Best Known (31, 31+11, s)-Nets in Base 2
(31, 31+11, 72)-Net over F2 — Constructive and digital
Digital (31, 42, 72)-net over F2, using
- trace code for nets [i] based on digital (3, 14, 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
(31, 31+11, 110)-Net over F2 — Digital
Digital (31, 42, 110)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(242, 110, F2, 2, 11) (dual of [(110, 2), 178, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(242, 132, F2, 2, 11) (dual of [(132, 2), 222, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(242, 264, F2, 11) (dual of [264, 222, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(242, 265, F2, 11) (dual of [265, 223, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(241, 256, F2, 11) (dual of [256, 215, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(233, 256, F2, 9) (dual of [256, 223, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 255 = 28−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(242, 265, F2, 11) (dual of [265, 223, 12]-code), using
- OOA 2-folding [i] based on linear OA(242, 264, F2, 11) (dual of [264, 222, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(242, 132, F2, 2, 11) (dual of [(132, 2), 222, 12]-NRT-code), using
(31, 31+11, 759)-Net in Base 2 — Upper bound on s
There is no (31, 42, 760)-net in base 2, because
- 1 times m-reduction [i] would yield (31, 41, 760)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2 211659 892903 > 241 [i]