Best Known (193−28, 193, s)-Nets in Base 2
(193−28, 193, 624)-Net over F2 — Constructive and digital
Digital (165, 193, 624)-net over F2, using
- 21 times duplication [i] based on digital (164, 192, 624)-net over F2, using
- t-expansion [i] based on digital (163, 192, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 32, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 32, 104)-net over F64, using
- t-expansion [i] based on digital (163, 192, 624)-net over F2, using
(193−28, 193, 2334)-Net over F2 — Digital
Digital (165, 193, 2334)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2193, 2334, F2, 3, 28) (dual of [(2334, 3), 6809, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2193, 2743, F2, 3, 28) (dual of [(2743, 3), 8036, 29]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2190, 2742, F2, 3, 28) (dual of [(2742, 3), 8036, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2190, 8226, F2, 28) (dual of [8226, 8036, 29]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2188, 8224, F2, 28) (dual of [8224, 8036, 29]-code), using
- 1 times truncation [i] based on linear OA(2189, 8225, F2, 29) (dual of [8225, 8036, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2183, 8193, F2, 29) (dual of [8193, 8010, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2157, 8193, F2, 25) (dual of [8193, 8036, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(2189, 8225, F2, 29) (dual of [8225, 8036, 30]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2188, 8224, F2, 28) (dual of [8224, 8036, 29]-code), using
- OOA 3-folding [i] based on linear OA(2190, 8226, F2, 28) (dual of [8226, 8036, 29]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2190, 2742, F2, 3, 28) (dual of [(2742, 3), 8036, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2193, 2743, F2, 3, 28) (dual of [(2743, 3), 8036, 29]-NRT-code), using
(193−28, 193, 85362)-Net in Base 2 — Upper bound on s
There is no (165, 193, 85363)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 12555 292490 139105 256662 821444 092031 615026 096243 210853 207532 > 2193 [i]