Best Known (172, 172+34, s)-Nets in Base 2
(172, 172+34, 490)-Net over F2 — Constructive and digital
Digital (172, 206, 490)-net over F2, using
- 21 times duplication [i] based on digital (171, 205, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 41, 98)-net over F32, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 7 and N(F) ≥ 98, using
- net from sequence [i] based on digital (7, 97)-sequence over F32, using
- trace code for nets [i] based on digital (7, 41, 98)-net over F32, using
(172, 172+34, 1268)-Net over F2 — Digital
Digital (172, 206, 1268)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2206, 1268, F2, 3, 34) (dual of [(1268, 3), 3598, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2206, 1369, F2, 3, 34) (dual of [(1369, 3), 3901, 35]-NRT-code), using
- strength reduction [i] based on linear OOA(2206, 1369, F2, 3, 35) (dual of [(1369, 3), 3901, 36]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2206, 4107, F2, 35) (dual of [4107, 3901, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2206, 4109, F2, 35) (dual of [4109, 3903, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(2205, 4096, F2, 35) (dual of [4096, 3891, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2193, 4096, F2, 33) (dual of [4096, 3903, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(34) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(2206, 4109, F2, 35) (dual of [4109, 3903, 36]-code), using
- OOA 3-folding [i] based on linear OA(2206, 4107, F2, 35) (dual of [4107, 3901, 36]-code), using
- strength reduction [i] based on linear OOA(2206, 1369, F2, 3, 35) (dual of [(1369, 3), 3901, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2206, 1369, F2, 3, 34) (dual of [(1369, 3), 3901, 35]-NRT-code), using
(172, 172+34, 31869)-Net in Base 2 — Upper bound on s
There is no (172, 206, 31870)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 102 852235 346708 791829 325286 970644 218666 411318 397890 184536 648063 > 2206 [i]