Best Known (138−25, 138, s)-Nets in Base 2
(138−25, 138, 266)-Net over F2 — Constructive and digital
Digital (113, 138, 266)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (2, 14, 6)-net over F2, using
- net from sequence [i] based on digital (2, 5)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 2 and N(F) ≥ 6, using
- Niederreiter–Xing sequence (Piršić implementation) with equidistant coordinate [i]
- net from sequence [i] based on digital (2, 5)-sequence over F2, using
- digital (99, 124, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 31, 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, 31, 65)-net over F16, using
- digital (2, 14, 6)-net over F2, using
(138−25, 138, 688)-Net over F2 — Digital
Digital (113, 138, 688)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2138, 688, F2, 3, 25) (dual of [(688, 3), 1926, 26]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2135, 687, F2, 3, 25) (dual of [(687, 3), 1926, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2135, 2061, F2, 25) (dual of [2061, 1926, 26]-code), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on linear OA(2134, 2060, F2, 25) (dual of [2060, 1926, 26]-code), using
- OOA 3-folding [i] based on linear OA(2135, 2061, F2, 25) (dual of [2061, 1926, 26]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2135, 687, F2, 3, 25) (dual of [(687, 3), 1926, 26]-NRT-code), using
(138−25, 138, 14440)-Net in Base 2 — Upper bound on s
There is no (113, 138, 14441)-net in base 2, because
- 1 times m-reduction [i] would yield (113, 137, 14441)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 174236 232116 943851 185763 756769 821928 688296 > 2137 [i]