Best Known (157−74, 157, s)-Nets in Base 2
(157−74, 157, 51)-Net over F2 — Constructive and digital
Digital (83, 157, 51)-net over F2, using
- t-expansion [i] based on digital (80, 157, 51)-net over F2, using
- net from sequence [i] based on digital (80, 50)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 2 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (80, 50)-sequence over F2, using
(157−74, 157, 57)-Net over F2 — Digital
Digital (83, 157, 57)-net over F2, using
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
(157−74, 157, 209)-Net over F2 — Upper bound on s (digital)
There is no digital (83, 157, 210)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2157, 210, F2, 74) (dual of [210, 53, 75]-code), but
- construction Y1 [i] would yield
- linear OA(2156, 190, F2, 74) (dual of [190, 34, 75]-code), but
- construction Y1 [i] would yield
- linear OA(2155, 178, F2, 74) (dual of [178, 23, 75]-code), but
- adding a parity check bit [i] would yield linear OA(2156, 179, F2, 75) (dual of [179, 23, 76]-code), but
- OA(234, 190, 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(2155, 178, F2, 74) (dual of [178, 23, 75]-code), but
- construction Y1 [i] would yield
- OA(253, 210, 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(2156, 190, F2, 74) (dual of [190, 34, 75]-code), but
- construction Y1 [i] would yield
(157−74, 157, 226)-Net in Base 2 — Upper bound on s
There is no (83, 157, 227)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 182838 986290 282005 319624 339884 844176 655617 617712 > 2157 [i]