Best Known (164−68, 164, s)-Nets in Base 2
(164−68, 164, 60)-Net over F2 — Constructive and digital
Digital (96, 164, 60)-net over F2, using
- 2 times m-reduction [i] based on digital (96, 166, 60)-net over F2, using
- trace code for nets [i] based on digital (13, 83, 30)-net over F4, using
- net from sequence [i] based on digital (13, 29)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 13 and N(F) ≥ 30, using
- F4 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 13 and N(F) ≥ 30, using
- net from sequence [i] based on digital (13, 29)-sequence over F4, using
- trace code for nets [i] based on digital (13, 83, 30)-net over F4, using
(164−68, 164, 72)-Net over F2 — Digital
Digital (96, 164, 72)-net over F2, using
(164−68, 164, 298)-Net in Base 2 — Upper bound on s
There is no (96, 164, 299)-net in base 2, because
- extracting embedded orthogonal array [i] would yield OA(2164, 299, S2, 68), but
- adding a parity check bit [i] would yield OA(2165, 300, S2, 69), but
- the linear programming bound shows that M ≥ 16470 634853 423617 222073 937683 004226 882121 322461 169983 603231 174966 959823 184323 923734 741973 421953 327317 020595 175640 350003 298665 312843 319464 957086 063230 716039 558211 565929 557659 891855 649331 547551 368093 958697 758454 972416 / 349 143115 130243 011384 433516 398585 633720 881828 307619 663705 534632 413548 149676 084959 032366 824299 505194 356375 466018 862585 539990 013735 677427 598317 259739 311972 598749 158217 > 2165 [i]
- adding a parity check bit [i] would yield OA(2165, 300, S2, 69), but