Best Known (226−37, 226, s)-Nets in Base 2
(226−37, 226, 490)-Net over F2 — Constructive and digital
Digital (189, 226, 490)-net over F2, using
- 21 times duplication [i] based on digital (188, 225, 490)-net over F2, using
- t-expansion [i] based on digital (187, 225, 490)-net over F2, using
- trace code for nets [i] based on digital (7, 45, 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, 45, 98)-net over F32, using
- t-expansion [i] based on digital (187, 225, 490)-net over F2, using
(226−37, 226, 1376)-Net over F2 — Digital
Digital (189, 226, 1376)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2226, 1376, F2, 3, 37) (dual of [(1376, 3), 3902, 38]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2224, 1376, F2, 3, 37) (dual of [(1376, 3), 3904, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2224, 4128, F2, 37) (dual of [4128, 3904, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2223, 4127, F2, 37) (dual of [4127, 3904, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(2217, 4097, F2, 37) (dual of [4097, 3880, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 224−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- 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(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([0,18]) ⊂ C([0,16]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2223, 4127, F2, 37) (dual of [4127, 3904, 38]-code), using
- OOA 3-folding [i] based on linear OA(2224, 4128, F2, 37) (dual of [4128, 3904, 38]-code), using
- 22 times duplication [i] based on linear OOA(2224, 1376, F2, 3, 37) (dual of [(1376, 3), 3904, 38]-NRT-code), using
(226−37, 226, 43726)-Net in Base 2 — Upper bound on s
There is no (189, 226, 43727)-net in base 2, because
- 1 times m-reduction [i] would yield (189, 225, 43727)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 53 937158 974880 437441 557214 692911 530968 370061 662858 182685 048854 592340 > 2225 [i]