Best Known (171, 171+37, s)-Nets in Base 2
(171, 171+37, 320)-Net over F2 — Constructive and digital
Digital (171, 208, 320)-net over F2, using
- 2 times m-reduction [i] based on digital (171, 210, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 42, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 42, 64)-net over F32, using
(171, 171+37, 854)-Net over F2 — Digital
Digital (171, 208, 854)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2208, 854, F2, 2, 37) (dual of [(854, 2), 1500, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2208, 1040, F2, 2, 37) (dual of [(1040, 2), 1872, 38]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2206, 1039, F2, 2, 37) (dual of [(1039, 2), 1872, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2206, 2078, F2, 37) (dual of [2078, 1872, 38]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2205, 2077, F2, 37) (dual of [2077, 1872, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,16]) [i] based on
- linear OA(2199, 2049, F2, 37) (dual of [2049, 1850, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(2177, 2049, F2, 33) (dual of [2049, 1872, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(26, 28, F2, 3) (dual of [28, 22, 4]-code or 28-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(2205, 2077, F2, 37) (dual of [2077, 1872, 38]-code), using
- OOA 2-folding [i] based on linear OA(2206, 2078, F2, 37) (dual of [2078, 1872, 38]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2206, 1039, F2, 2, 37) (dual of [(1039, 2), 1872, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2208, 1040, F2, 2, 37) (dual of [(1040, 2), 1872, 38]-NRT-code), using
(171, 171+37, 21849)-Net in Base 2 — Upper bound on s
There is no (171, 208, 21850)-net in base 2, because
- 1 times m-reduction [i] would yield (171, 207, 21850)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 205 709761 585722 323447 344355 849038 397522 952726 645544 565155 285156 > 2207 [i]