Best Known (82, 82+93, s)-Nets in Base 2
(82, 82+93, 51)-Net over F2 — Constructive and digital
Digital (82, 175, 51)-net over F2, using
- t-expansion [i] based on digital (80, 175, 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+93, 56)-Net over F2 — Digital
Digital (82, 175, 56)-net over F2, using
- t-expansion [i] based on digital (80, 175, 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+93, 174)-Net over F2 — Upper bound on s (digital)
There is no digital (82, 175, 175)-net over F2, because
- 9 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+93, 175)-Net in Base 2 — Upper bound on s
There is no (82, 175, 176)-net in base 2, because
- 5 times m-reduction [i] would yield (82, 170, 176)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2170, 176, S2, 88), but
- adding a parity check bit [i] would yield OA(2171, 177, S2, 89), but
- the (dual) Plotkin bound shows that M ≥ 47890 485652 059026 823698 344598 447161 988085 597568 237568 / 15 > 2171 [i]
- adding a parity check bit [i] would yield OA(2171, 177, S2, 89), but
- extracting embedded orthogonal array [i] would yield OA(2170, 176, S2, 88), but