Best Known (164, 198, s)-Nets in Base 2
(164, 198, 380)-Net over F2 — Constructive and digital
Digital (164, 198, 380)-net over F2, using
- 23 times duplication [i] based on digital (161, 195, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 39, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 39, 76)-net over F32, using
(164, 198, 949)-Net over F2 — Digital
Digital (164, 198, 949)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2198, 949, F2, 2, 34) (dual of [(949, 2), 1700, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2198, 1042, F2, 2, 34) (dual of [(1042, 2), 1886, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2198, 2084, F2, 34) (dual of [2084, 1886, 35]-code), using
- 1 times truncation [i] based on linear OA(2199, 2085, F2, 35) (dual of [2085, 1886, 36]-code), using
- construction XX applied to Ce(34) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2188, 2048, F2, 35) (dual of [2048, 1860, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2166, 2048, F2, 31) (dual of [2048, 1882, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- linear OA(21, 5, F2, 1) (dual of [5, 4, 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 XX applied to Ce(34) ⊂ Ce(30) ⊂ Ce(28) [i] based on
- 1 times truncation [i] based on linear OA(2199, 2085, F2, 35) (dual of [2085, 1886, 36]-code), using
- OOA 2-folding [i] based on linear OA(2198, 2084, F2, 34) (dual of [2084, 1886, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(2198, 1042, F2, 2, 34) (dual of [(1042, 2), 1886, 35]-NRT-code), using
(164, 198, 22992)-Net in Base 2 — Upper bound on s
There is no (164, 198, 22993)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 401861 579802 375674 138476 421584 636301 369022 634839 842427 452874 > 2198 [i]