Best Known (82, 82+87, s)-Nets in Base 2
(82, 82+87, 51)-Net over F2 — Constructive and digital
Digital (82, 169, 51)-net over F2, using
- t-expansion [i] based on digital (80, 169, 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
(82, 82+87, 56)-Net over F2 — Digital
Digital (82, 169, 56)-net over F2, using
- t-expansion [i] based on digital (80, 169, 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
(82, 82+87, 174)-Net over F2 — Upper bound on s (digital)
There is no digital (82, 169, 175)-net over F2, because
- 3 times m-reduction [i] would yield digital (82, 166, 175)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2166, 175, F2, 84) (dual of [175, 9, 85]-code), but
- residual code [i] would yield linear OA(282, 90, F2, 42) (dual of [90, 8, 43]-code), but
- adding a parity check bit [i] would yield linear OA(283, 91, F2, 43) (dual of [91, 8, 44]-code), but
- “DMa†bound on codes from Brouwer’s database [i]
- adding a parity check bit [i] would yield linear OA(283, 91, F2, 43) (dual of [91, 8, 44]-code), but
- residual code [i] would yield linear OA(282, 90, F2, 42) (dual of [90, 8, 43]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2166, 175, F2, 84) (dual of [175, 9, 85]-code), but
(82, 82+87, 195)-Net in Base 2 — Upper bound on s
There is no (82, 169, 196)-net in base 2, because
- 1 times m-reduction [i] would yield (82, 168, 196)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 388 657169 590100 544444 101435 474035 328787 030851 408232 > 2168 [i]