Best Known (127−65, 127, s)-Nets in Base 2
(127−65, 127, 43)-Net over F2 — Constructive and digital
Digital (62, 127, 43)-net over F2, using
- t-expansion [i] based on digital (59, 127, 43)-net over F2, using
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 54, N(F) = 42, and 1 place with degree 6 [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (59, 42)-sequence over F2, using
(127−65, 127, 44)-Net over F2 — Digital
Digital (62, 127, 44)-net over F2, using
- net from sequence [i] based on digital (62, 43)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 62 and N(F) ≥ 44, using
(127−65, 127, 132)-Net over F2 — Upper bound on s (digital)
There is no digital (62, 127, 133)-net over F2, because
- 1 times m-reduction [i] would yield digital (62, 126, 133)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2126, 133, F2, 64) (dual of [133, 7, 65]-code), but
(127−65, 127, 135)-Net in Base 2 — Upper bound on s
There is no (62, 127, 136)-net in base 2, because
- 3 times m-reduction [i] would yield (62, 124, 136)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2124, 136, S2, 62), but
- the linear programming bound shows that M ≥ 17949 894855 079503 947693 010542 025773 154304 / 703 > 2124 [i]
- extracting embedded orthogonal array [i] would yield OA(2124, 136, S2, 62), but