Best Known (87, 167, s)-Nets in Base 2
(87, 167, 52)-Net over F2 — Constructive and digital
Digital (87, 167, 52)-net over F2, using
- t-expansion [i] based on digital (85, 167, 52)-net over F2, using
- net from sequence [i] based on digital (85, 51)-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 3 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 (85, 51)-sequence over F2, using
(87, 167, 57)-Net over F2 — Digital
Digital (87, 167, 57)-net over F2, using
- t-expansion [i] based on digital (83, 167, 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
(87, 167, 193)-Net over F2 — Upper bound on s (digital)
There is no digital (87, 167, 194)-net over F2, because
- 2 times m-reduction [i] would yield digital (87, 165, 194)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2165, 194, F2, 78) (dual of [194, 29, 79]-code), but
- 1 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
- 1 times code embedding in larger space [i] would yield linear OA(2166, 195, F2, 78) (dual of [195, 29, 79]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2165, 194, F2, 78) (dual of [194, 29, 79]-code), but
(87, 167, 230)-Net in Base 2 — Upper bound on s
There is no (87, 167, 231)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 202 665893 558122 684444 824539 642393 632274 884971 394200 > 2167 [i]