Best Known (177−31, 177, s)-Nets in Base 2
(177−31, 177, 320)-Net over F2 — Constructive and digital
Digital (146, 177, 320)-net over F2, using
- 22 times duplication [i] based on digital (144, 175, 320)-net over F2, using
- t-expansion [i] based on digital (143, 175, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 35, 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, 35, 64)-net over F32, using
- t-expansion [i] based on digital (143, 175, 320)-net over F2, using
(177−31, 177, 820)-Net over F2 — Digital
Digital (146, 177, 820)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2177, 820, F2, 2, 31) (dual of [(820, 2), 1463, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2177, 1042, F2, 2, 31) (dual of [(1042, 2), 1907, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2177, 2084, F2, 31) (dual of [2084, 1907, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2177, 2085, F2, 31) (dual of [2085, 1908, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(26) ⊂ Ce(24) [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(2133, 2048, F2, 25) (dual of [2048, 1915, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(30) ⊂ Ce(26) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2177, 2085, F2, 31) (dual of [2085, 1908, 32]-code), using
- OOA 2-folding [i] based on linear OA(2177, 2084, F2, 31) (dual of [2084, 1907, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(2177, 1042, F2, 2, 31) (dual of [(1042, 2), 1907, 32]-NRT-code), using
(177−31, 177, 21848)-Net in Base 2 — Upper bound on s
There is no (146, 177, 21849)-net in base 2, because
- 1 times m-reduction [i] would yield (146, 176, 21849)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 95835 632239 073289 814016 276312 298477 120100 938421 186816 > 2176 [i]