Best Known (172−28, 172, s)-Nets in Base 2
(172−28, 172, 390)-Net over F2 — Constructive and digital
Digital (144, 172, 390)-net over F2, using
- 24 times duplication [i] based on digital (140, 168, 390)-net over F2, using
- trace code for nets [i] based on digital (0, 28, 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, 28, 65)-net over F64, using
(172−28, 172, 1257)-Net over F2 — Digital
Digital (144, 172, 1257)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2172, 1257, F2, 3, 28) (dual of [(1257, 3), 3599, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2172, 1370, F2, 3, 28) (dual of [(1370, 3), 3938, 29]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2171, 1370, F2, 3, 28) (dual of [(1370, 3), 3939, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2171, 4110, F2, 28) (dual of [4110, 3939, 29]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2169, 4108, F2, 28) (dual of [4108, 3939, 29]-code), using
- 1 times truncation [i] based on linear OA(2170, 4109, F2, 29) (dual of [4109, 3939, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2169, 4096, F2, 29) (dual of [4096, 3927, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2157, 4096, F2, 27) (dual of [4096, 3939, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(28) ⊂ Ce(26) [i] based on
- 1 times truncation [i] based on linear OA(2170, 4109, F2, 29) (dual of [4109, 3939, 30]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2169, 4108, F2, 28) (dual of [4108, 3939, 29]-code), using
- OOA 3-folding [i] based on linear OA(2171, 4110, F2, 28) (dual of [4110, 3939, 29]-code), using
- 21 times duplication [i] based on linear OOA(2171, 1370, F2, 3, 28) (dual of [(1370, 3), 3939, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2172, 1370, F2, 3, 28) (dual of [(1370, 3), 3938, 29]-NRT-code), using
(172−28, 172, 30166)-Net in Base 2 — Upper bound on s
There is no (144, 172, 30167)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 5986 465075 783540 028044 317082 991924 670727 477447 057162 > 2172 [i]