Best Known (109, 109+25, s)-Nets in Base 2
(109, 109+25, 260)-Net over F2 — Constructive and digital
Digital (109, 134, 260)-net over F2, using
- t-expansion [i] based on digital (108, 134, 260)-net over F2, using
- 2 times m-reduction [i] based on digital (108, 136, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 34, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 34, 65)-net over F16, using
- 2 times m-reduction [i] based on digital (108, 136, 260)-net over F2, using
(109, 109+25, 625)-Net over F2 — Digital
Digital (109, 134, 625)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2134, 625, F2, 3, 25) (dual of [(625, 3), 1741, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2134, 686, F2, 3, 25) (dual of [(686, 3), 1924, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2134, 2058, F2, 25) (dual of [2058, 1924, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2134, 2060, F2, 25) (dual of [2060, 1926, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(2133, 2048, F2, 25) (dual of [2048, 1915, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(2122, 2048, F2, 23) (dual of [2048, 1926, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(2134, 2060, F2, 25) (dual of [2060, 1926, 26]-code), using
- OOA 3-folding [i] based on linear OA(2134, 2058, F2, 25) (dual of [2058, 1924, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(2134, 686, F2, 3, 25) (dual of [(686, 3), 1924, 26]-NRT-code), using
(109, 109+25, 11458)-Net in Base 2 — Upper bound on s
There is no (109, 134, 11459)-net in base 2, because
- 1 times m-reduction [i] would yield (109, 133, 11459)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 10898 610050 701868 281971 866840 701764 364752 > 2133 [i]