Best Known (148, 177, s)-Nets in Base 2
(148, 177, 390)-Net over F2 — Constructive and digital
Digital (148, 177, 390)-net over F2, using
- 23 times duplication [i] based on digital (145, 174, 390)-net over F2, using
- trace code for nets [i] based on digital (0, 29, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- trace code for nets [i] based on digital (0, 29, 65)-net over F64, using
(148, 177, 1231)-Net over F2 — Digital
Digital (148, 177, 1231)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2177, 1231, F2, 3, 29) (dual of [(1231, 3), 3516, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2177, 1376, F2, 3, 29) (dual of [(1376, 3), 3951, 30]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2176, 1376, F2, 3, 29) (dual of [(1376, 3), 3952, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2176, 4128, F2, 29) (dual of [4128, 3952, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2175, 4127, F2, 29) (dual of [4127, 3952, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2169, 4097, F2, 29) (dual of [4097, 3928, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2145, 4097, F2, 25) (dual of [4097, 3952, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(26, 30, F2, 3) (dual of [30, 24, 4]-code or 30-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 C([0,14]) ⊂ C([0,12]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2175, 4127, F2, 29) (dual of [4127, 3952, 30]-code), using
- OOA 3-folding [i] based on linear OA(2176, 4128, F2, 29) (dual of [4128, 3952, 30]-code), using
- 21 times duplication [i] based on linear OOA(2176, 1376, F2, 3, 29) (dual of [(1376, 3), 3952, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2177, 1376, F2, 3, 29) (dual of [(1376, 3), 3951, 30]-NRT-code), using
(148, 177, 36778)-Net in Base 2 — Upper bound on s
There is no (148, 177, 36779)-net in base 2, because
- 1 times m-reduction [i] would yield (148, 176, 36779)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 95802 677088 774243 218922 344645 777845 544075 928369 875688 > 2176 [i]