Best Known (91, 91+93, s)-Nets in Base 2
(91, 91+93, 53)-Net over F2 — Constructive and digital
Digital (91, 184, 53)-net over F2, using
- t-expansion [i] based on digital (90, 184, 53)-net over F2, using
- net from sequence [i] based on digital (90, 52)-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 4 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 (90, 52)-sequence over F2, using
(91, 91+93, 57)-Net over F2 — Digital
Digital (91, 184, 57)-net over F2, using
- t-expansion [i] based on digital (83, 184, 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
- net from sequence [i] based on digital (83, 56)-sequence over F2, using
(91, 91+93, 193)-Net over F2 — Upper bound on s (digital)
There is no digital (91, 184, 194)-net over F2, because
- 1 times m-reduction [i] would yield digital (91, 183, 194)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2183, 194, F2, 92) (dual of [194, 11, 93]-code), but
- residual code [i] would yield linear OA(291, 101, F2, 46) (dual of [101, 10, 47]-code), but
- adding a parity check bit [i] would yield linear OA(292, 102, F2, 47) (dual of [102, 10, 48]-code), but
- “Bro†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(292, 102, F2, 47) (dual of [102, 10, 48]-code), but
- residual code [i] would yield linear OA(291, 101, F2, 46) (dual of [101, 10, 47]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2183, 194, F2, 92) (dual of [194, 11, 93]-code), but
(91, 91+93, 221)-Net in Base 2 — Upper bound on s
There is no (91, 184, 222)-net in base 2, because
- 1 times m-reduction [i] would yield (91, 183, 222)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 12 550316 543411 138365 102610 545369 698662 940881 687983 798720 > 2183 [i]