Best Known (214−93, 214, s)-Nets in Base 2
(214−93, 214, 63)-Net over F2 — Constructive and digital
Digital (121, 214, 63)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (21, 67, 21)-net over F2, using
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 21 and N(F) ≥ 21, using
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
- digital (54, 147, 42)-net over F2, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- digital (21, 67, 21)-net over F2, using
(214−93, 214, 81)-Net over F2 — Digital
Digital (121, 214, 81)-net over F2, using
(214−93, 214, 357)-Net over F2 — Upper bound on s (digital)
There is no digital (121, 214, 358)-net over F2, because
- 1 times m-reduction [i] would yield digital (121, 213, 358)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2213, 358, F2, 92) (dual of [358, 145, 93]-code), but
- construction Y1 [i] would yield
- OA(2212, 300, S2, 92), but
- the linear programming bound shows that M ≥ 2484 132694 351399 471861 447697 940992 402712 231248 004226 234322 576635 442083 069406 167011 327832 607273 517056 / 377277 818302 855799 777736 052112 623275 > 2212 [i]
- linear OA(2145, 358, F2, 58) (dual of [358, 213, 59]-code), but
- discarding factors / shortening the dual code would yield linear OA(2145, 343, F2, 58) (dual of [343, 198, 59]-code), but
- the improved Johnson bound shows that N ≤ 7 839683 395577 661521 065550 956096 419139 602118 236678 534951 091663 < 2198 [i]
- discarding factors / shortening the dual code would yield linear OA(2145, 343, F2, 58) (dual of [343, 198, 59]-code), but
- OA(2212, 300, S2, 92), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2213, 358, F2, 92) (dual of [358, 145, 93]-code), but
(214−93, 214, 381)-Net in Base 2 — Upper bound on s
There is no (121, 214, 382)-net in base 2, because
- 1 times m-reduction [i] would yield (121, 213, 382)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 13534 903022 146747 716873 453686 164488 852613 300268 674865 197654 217648 > 2213 [i]