Best Known (173−31, 173, s)-Nets in Base 2
(173−31, 173, 320)-Net over F2 — Constructive and digital
Digital (142, 173, 320)-net over F2, using
- 23 times duplication [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
(173−31, 173, 739)-Net over F2 — Digital
Digital (142, 173, 739)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2173, 739, F2, 2, 31) (dual of [(739, 2), 1305, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2173, 1038, F2, 2, 31) (dual of [(1038, 2), 1903, 32]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2172, 1038, F2, 2, 31) (dual of [(1038, 2), 1904, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2172, 2076, F2, 31) (dual of [2076, 1904, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- 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(2144, 2048, F2, 27) (dual of [2048, 1904, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-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(30) ⊂ Ce(26) [i] based on
- OOA 2-folding [i] based on linear OA(2172, 2076, F2, 31) (dual of [2076, 1904, 32]-code), using
- 21 times duplication [i] based on linear OOA(2172, 1038, F2, 2, 31) (dual of [(1038, 2), 1904, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2173, 1038, F2, 2, 31) (dual of [(1038, 2), 1903, 32]-NRT-code), using
(173−31, 173, 18157)-Net in Base 2 — Upper bound on s
There is no (142, 173, 18158)-net in base 2, because
- 1 times m-reduction [i] would yield (142, 172, 18158)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 5989 688706 975820 751890 895531 951395 123066 190718 967260 > 2172 [i]