Best Known (191, 191+35, s)-Nets in Base 2
(191, 191+35, 520)-Net over F2 — Constructive and digital
Digital (191, 226, 520)-net over F2, using
- 21 times duplication [i] based on digital (190, 225, 520)-net over F2, using
- t-expansion [i] based on digital (189, 225, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 45, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- trace code for nets [i] based on digital (9, 45, 104)-net over F32, using
- t-expansion [i] based on digital (189, 225, 520)-net over F2, using
(191, 191+35, 1991)-Net over F2 — Digital
Digital (191, 226, 1991)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2226, 1991, F2, 4, 35) (dual of [(1991, 4), 7738, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2226, 2052, F2, 4, 35) (dual of [(2052, 4), 7982, 36]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2225, 2052, F2, 4, 35) (dual of [(2052, 4), 7983, 36]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2225, 8208, F2, 35) (dual of [8208, 7983, 36]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2223, 8206, F2, 35) (dual of [8206, 7983, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(2222, 8192, F2, 35) (dual of [8192, 7970, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2209, 8192, F2, 33) (dual of [8192, 7983, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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(34) ⊂ Ce(32) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2223, 8206, F2, 35) (dual of [8206, 7983, 36]-code), using
- OOA 4-folding [i] based on linear OA(2225, 8208, F2, 35) (dual of [8208, 7983, 36]-code), using
- 21 times duplication [i] based on linear OOA(2225, 2052, F2, 4, 35) (dual of [(2052, 4), 7983, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2226, 2052, F2, 4, 35) (dual of [(2052, 4), 7982, 36]-NRT-code), using
(191, 191+35, 69184)-Net in Base 2 — Upper bound on s
There is no (191, 226, 69185)-net in base 2, because
- 1 times m-reduction [i] would yield (191, 225, 69185)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 53 926406 893144 348816 854339 372487 414061 135043 340591 413604 664538 183480 > 2225 [i]