Best Known (153−73, 153, s)-Nets in Base 2
(153−73, 153, 51)-Net over F2 — Constructive and digital
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]
(153−73, 153, 56)-Net over F2 — Digital
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
(153−73, 153, 178)-Net over F2 — Upper bound on s (digital)
There is no digital (80, 153, 179)-net over F2, because
- 1 times m-reduction [i] would yield digital (80, 152, 179)-net over F2, but
- extracting embedded orthogonal array [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
- extracting embedded orthogonal array [i] would yield linear OA(2152, 179, F2, 72) (dual of [179, 27, 73]-code), but
(153−73, 153, 217)-Net in Base 2 — Upper bound on s
There is no (80, 153, 218)-net in base 2, because
- 1 times m-reduction [i] would yield (80, 152, 218)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6147 307799 497874 221549 205729 177911 519423 790780 > 2152 [i]