Best Known (168−24, 168, s)-Nets in Base 2
(168−24, 168, 1365)-Net over F2 — Constructive and digital
Digital (144, 168, 1365)-net over F2, using
- net defined by OOA [i] based on linear OOA(2168, 1365, F2, 24, 24) (dual of [(1365, 24), 32592, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(2168, 16380, F2, 24) (dual of [16380, 16212, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2168, 16384, F2, 24) (dual of [16384, 16216, 25]-code), using
- 1 times truncation [i] based on linear OA(2169, 16385, F2, 25) (dual of [16385, 16216, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(2169, 16385, F2, 25) (dual of [16385, 16216, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2168, 16384, F2, 24) (dual of [16384, 16216, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(2168, 16380, F2, 24) (dual of [16380, 16212, 25]-code), using
(168−24, 168, 3276)-Net over F2 — Digital
Digital (144, 168, 3276)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2168, 3276, F2, 5, 24) (dual of [(3276, 5), 16212, 25]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2168, 16380, F2, 24) (dual of [16380, 16212, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2168, 16384, F2, 24) (dual of [16384, 16216, 25]-code), using
- 1 times truncation [i] based on linear OA(2169, 16385, F2, 25) (dual of [16385, 16216, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16385 | 228−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(2169, 16385, F2, 25) (dual of [16385, 16216, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2168, 16384, F2, 24) (dual of [16384, 16216, 25]-code), using
- OOA 5-folding [i] based on linear OA(2168, 16380, F2, 24) (dual of [16380, 16212, 25]-code), using
(168−24, 168, 86635)-Net in Base 2 — Upper bound on s
There is no (144, 168, 86636)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 374 193506 514460 921111 746940 748267 753390 058287 244392 > 2168 [i]