Best Known (82, 82+78, s)-Nets in Base 2
(82, 82+78, 51)-Net over F2 — Constructive and digital
Digital (82, 160, 51)-net over F2, using
- t-expansion [i] based on digital (80, 160, 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+78, 56)-Net over F2 — Digital
Digital (82, 160, 56)-net over F2, using
- t-expansion [i] based on digital (80, 160, 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+78, 188)-Net over F2 — Upper bound on s (digital)
There is no digital (82, 160, 189)-net over F2, because
- extracting embedded orthogonal array [i] would yield linear OA(2160, 189, F2, 78) (dual of [189, 29, 79]-code), but
- 6 times code embedding in larger space [i] would yield linear OA(2166, 195, F2, 78) (dual of [195, 29, 79]-code), but
- adding a parity check bit [i] would yield linear OA(2167, 196, F2, 79) (dual of [196, 29, 80]-code), but
- 6 times code embedding in larger space [i] would yield linear OA(2166, 195, F2, 78) (dual of [195, 29, 79]-code), but
(82, 82+78, 211)-Net in Base 2 — Upper bound on s
There is no (82, 160, 212)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1 530096 191591 810865 400299 649862 196970 817145 371640 > 2160 [i]