Best Known (139, 168, s)-Nets in Base 2
(139, 168, 320)-Net over F2 — Constructive and digital
Digital (139, 168, 320)-net over F2, using
- 2 times m-reduction [i] based on digital (139, 170, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 34, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 34, 64)-net over F32, using
(139, 168, 844)-Net over F2 — Digital
Digital (139, 168, 844)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2168, 844, F2, 2, 29) (dual of [(844, 2), 1520, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2168, 1047, F2, 2, 29) (dual of [(1047, 2), 1926, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2168, 2094, F2, 29) (dual of [2094, 1926, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(2155, 2048, F2, 29) (dual of [2048, 1893, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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(213, 46, F2, 5) (dual of [46, 33, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(2168, 2094, F2, 29) (dual of [2094, 1926, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2168, 1047, F2, 2, 29) (dual of [(1047, 2), 1926, 30]-NRT-code), using
(139, 168, 23547)-Net in Base 2 — Upper bound on s
There is no (139, 168, 23548)-net in base 2, because
- 1 times m-reduction [i] would yield (139, 167, 23548)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 187 170228 500605 045397 392196 609717 639134 720263 414564 > 2167 [i]