Best Known (178−28, 178, s)-Nets in Base 2
(178−28, 178, 490)-Net over F2 — Constructive and digital
Digital (150, 178, 490)-net over F2, using
- 23 times duplication [i] based on digital (147, 175, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 35, 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, 35, 98)-net over F32, using
(178−28, 178, 1376)-Net over F2 — Digital
Digital (150, 178, 1376)-net over F2, using
- 22 times duplication [i] based on digital (148, 176, 1376)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2176, 1376, F2, 3, 28) (dual of [(1376, 3), 3952, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2176, 4128, F2, 28) (dual of [4128, 3952, 29]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2174, 4126, F2, 28) (dual of [4126, 3952, 29]-code), using
- 1 times truncation [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 truncation [i] based on linear OA(2175, 4127, F2, 29) (dual of [4127, 3952, 30]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2174, 4126, F2, 28) (dual of [4126, 3952, 29]-code), using
- OOA 3-folding [i] based on linear OA(2176, 4128, F2, 28) (dual of [4128, 3952, 29]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2176, 1376, F2, 3, 28) (dual of [(1376, 3), 3952, 29]-NRT-code), using
(178−28, 178, 40608)-Net in Base 2 — Upper bound on s
There is no (150, 178, 40609)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 383152 476443 660249 001848 186608 886556 091075 870928 947696 > 2178 [i]