Best Known (132−62, 132, s)-Nets in Base 2
(132−62, 132, 49)-Net over F2 — Constructive and digital
Digital (70, 132, 49)-net over F2, using
- net from sequence [i] based on digital (70, 48)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, and 1 place with degree 2 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
(132−62, 132, 184)-Net over F2 — Upper bound on s (digital)
There is no digital (70, 132, 185)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2132, 185, F2, 62) (dual of [185, 53, 63]-code), but
- construction Y1 [i] would yield
- linear OA(2131, 165, F2, 62) (dual of [165, 34, 63]-code), but
- construction Y1 [i] would yield
- linear OA(2130, 153, F2, 62) (dual of [153, 23, 63]-code), but
- adding a parity check bit [i] would yield linear OA(2131, 154, F2, 63) (dual of [154, 23, 64]-code), but
- OA(234, 165, S2, 12), but
- discarding factors would yield OA(234, 154, S2, 12), but
- the Rao or (dual) Hamming bound shows that M ≥ 17486 314616 > 234 [i]
- discarding factors would yield OA(234, 154, S2, 12), but
- linear OA(2130, 153, F2, 62) (dual of [153, 23, 63]-code), but
- construction Y1 [i] would yield
- OA(253, 185, S2, 20), but
- discarding factors would yield OA(253, 182, S2, 20), but
- the Rao or (dual) Hamming bound shows that M ≥ 9064 436853 738748 > 253 [i]
- discarding factors would yield OA(253, 182, S2, 20), but
- linear OA(2131, 165, F2, 62) (dual of [165, 34, 63]-code), but
- construction Y1 [i] would yield
(132−62, 132, 195)-Net in Base 2 — Upper bound on s
There is no (70, 132, 196)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 5991 032047 578067 338414 077344 762501 214448 > 2132 [i]