Best Known (137−22, 137, s)-Nets in Base 2
(137−22, 137, 390)-Net over F2 — Constructive and digital
Digital (115, 137, 390)-net over F2, using
- 1 times m-reduction [i] based on digital (115, 138, 390)-net over F2, using
- trace code for nets [i] based on digital (0, 23, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 23, 65)-net over F64, using
(137−22, 137, 1290)-Net over F2 — Digital
Digital (115, 137, 1290)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2137, 1290, F2, 3, 22) (dual of [(1290, 3), 3733, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2137, 1370, F2, 3, 22) (dual of [(1370, 3), 3973, 23]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2135, 1370, F2, 3, 22) (dual of [(1370, 3), 3975, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2135, 4110, F2, 22) (dual of [4110, 3975, 23]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2133, 4108, F2, 22) (dual of [4108, 3975, 23]-code), using
- 1 times truncation [i] based on linear OA(2134, 4109, F2, 23) (dual of [4109, 3975, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2133, 4096, F2, 23) (dual of [4096, 3963, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2121, 4096, F2, 21) (dual of [4096, 3975, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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(22) ⊂ Ce(20) [i] based on
- 1 times truncation [i] based on linear OA(2134, 4109, F2, 23) (dual of [4109, 3975, 24]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2133, 4108, F2, 22) (dual of [4108, 3975, 23]-code), using
- OOA 3-folding [i] based on linear OA(2135, 4110, F2, 22) (dual of [4110, 3975, 23]-code), using
- 22 times duplication [i] based on linear OOA(2135, 1370, F2, 3, 22) (dual of [(1370, 3), 3975, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2137, 1370, F2, 3, 22) (dual of [(1370, 3), 3973, 23]-NRT-code), using
(137−22, 137, 27539)-Net in Base 2 — Upper bound on s
There is no (115, 137, 27540)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 174269 544304 425992 760905 208177 439682 784178 > 2137 [i]