Best Known (168, 168+33, s)-Nets in Base 2
(168, 168+33, 490)-Net over F2 — Constructive and digital
Digital (168, 201, 490)-net over F2, using
- 21 times duplication [i] based on digital (167, 200, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 40, 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, 40, 98)-net over F32, using
(168, 168+33, 1284)-Net over F2 — Digital
Digital (168, 201, 1284)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2201, 1284, F2, 3, 33) (dual of [(1284, 3), 3651, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2201, 1376, F2, 3, 33) (dual of [(1376, 3), 3927, 34]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2200, 1376, F2, 3, 33) (dual of [(1376, 3), 3928, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2200, 4128, F2, 33) (dual of [4128, 3928, 34]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2199, 4127, F2, 33) (dual of [4127, 3928, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,14]) [i] based on
- linear OA(2193, 4097, F2, 33) (dual of [4097, 3904, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 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(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([0,16]) ⊂ C([0,14]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2199, 4127, F2, 33) (dual of [4127, 3928, 34]-code), using
- OOA 3-folding [i] based on linear OA(2200, 4128, F2, 33) (dual of [4128, 3928, 34]-code), using
- 21 times duplication [i] based on linear OOA(2200, 1376, F2, 3, 33) (dual of [(1376, 3), 3928, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2201, 1376, F2, 3, 33) (dual of [(1376, 3), 3927, 34]-NRT-code), using
(168, 168+33, 39369)-Net in Base 2 — Upper bound on s
There is no (168, 201, 39370)-net in base 2, because
- 1 times m-reduction [i] would yield (168, 200, 39370)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 607579 198524 539952 688424 335801 663805 803296 447840 888720 627628 > 2200 [i]