Best Known (135−74, 135, s)-Nets in Base 2
(135−74, 135, 43)-Net over F2 — Constructive and digital
Digital (61, 135, 43)-net over F2, using
- t-expansion [i] based on digital (59, 135, 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
(135−74, 135, 130)-Net over F2 — Upper bound on s (digital)
There is no digital (61, 135, 131)-net over F2, because
- 10 times m-reduction [i] would yield digital (61, 125, 131)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2125, 131, F2, 64) (dual of [131, 6, 65]-code), but
(135−74, 135, 133)-Net in Base 2 — Upper bound on s
There is no (61, 135, 134)-net in base 2, because
- 12 times m-reduction [i] would yield (61, 123, 134)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2123, 134, S2, 62), but
- the linear programming bound shows that M ≥ 2988 104534 524490 882287 758271 510214 606848 / 247 > 2123 [i]
- extracting embedded orthogonal array [i] would yield OA(2123, 134, S2, 62), but