Best Known (138, 172, s)-Nets in Base 2
(138, 172, 260)-Net over F2 — Constructive and digital
Digital (138, 172, 260)-net over F2, using
- 4 times m-reduction [i] based on digital (138, 176, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 44, 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, 44, 65)-net over F16, using
(138, 172, 512)-Net over F2 — Digital
Digital (138, 172, 512)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2172, 512, F2, 2, 34) (dual of [(512, 2), 852, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2172, 522, F2, 2, 34) (dual of [(522, 2), 872, 35]-NRT-code), using
- strength reduction [i] based on linear OOA(2172, 522, F2, 2, 35) (dual of [(522, 2), 872, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2172, 1044, F2, 35) (dual of [1044, 872, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2172, 1045, F2, 35) (dual of [1045, 873, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(2166, 1024, F2, 35) (dual of [1024, 858, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2151, 1024, F2, 31) (dual of [1024, 873, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(26, 21, F2, 3) (dual of [21, 15, 4]-code or 21-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(2172, 1045, F2, 35) (dual of [1045, 873, 36]-code), using
- OOA 2-folding [i] based on linear OA(2172, 1044, F2, 35) (dual of [1044, 872, 36]-code), using
- strength reduction [i] based on linear OOA(2172, 522, F2, 2, 35) (dual of [(522, 2), 872, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2172, 522, F2, 2, 34) (dual of [(522, 2), 872, 35]-NRT-code), using
(138, 172, 7948)-Net in Base 2 — Upper bound on s
There is no (138, 172, 7949)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 5989 554214 923220 993196 066237 289773 608972 568529 880224 > 2172 [i]