Best Known (225−35, 225, s)-Nets in Base 2
(225−35, 225, 520)-Net over F2 — Constructive and digital
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
(225−35, 225, 1944)-Net over F2 — Digital
Digital (190, 225, 1944)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2225, 1944, F2, 4, 35) (dual of [(1944, 4), 7551, 36]-NRT-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OOA(2225, 2052, F2, 4, 35) (dual of [(2052, 4), 7983, 36]-NRT-code), using
(225−35, 225, 66419)-Net in Base 2 — Upper bound on s
There is no (190, 225, 66420)-net in base 2, because
- 1 times m-reduction [i] would yield (190, 224, 66420)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 26 964365 660107 913634 228929 117356 654395 688139 749364 781692 720280 467743 > 2224 [i]