Best Known (153−72, 153, s)-Nets in Base 2
(153−72, 153, 51)-Net over F2 — Constructive and digital
Digital (81, 153, 51)-net over F2, using
- t-expansion [i] based on digital (80, 153, 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
(153−72, 153, 56)-Net over F2 — Digital
Digital (81, 153, 56)-net over F2, using
- t-expansion [i] based on digital (80, 153, 56)-net over F2, using
- net from sequence [i] based on digital (80, 55)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 80 and N(F) ≥ 56, using
- net from sequence [i] based on digital (80, 55)-sequence over F2, using
(153−72, 153, 192)-Net over F2 — Upper bound on s (digital)
There is no digital (81, 153, 193)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2153, 193, F2, 72) (dual of [193, 40, 73]-code), but
- construction Y1 [i] would yield
- linear OA(2152, 179, F2, 72) (dual of [179, 27, 73]-code), but
- adding a parity check bit [i] would yield linear OA(2153, 180, F2, 73) (dual of [180, 27, 74]-code), but
- OA(240, 193, S2, 14), but
- discarding factors would yield OA(240, 180, S2, 14), but
- the Rao or (dual) Hamming bound shows that M ≥ 1 124371 299892 > 240 [i]
- discarding factors would yield OA(240, 180, S2, 14), but
- linear OA(2152, 179, F2, 72) (dual of [179, 27, 73]-code), but
- construction Y1 [i] would yield
(153−72, 153, 222)-Net in Base 2 — Upper bound on s
There is no (81, 153, 223)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 12104 839174 155941 172420 020347 626097 811916 310310 > 2153 [i]