Best Known (165−85, 165, s)-Nets in Base 2
(165−85, 165, 51)-Net over F2 — Constructive and digital
Digital (80, 165, 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]
(165−85, 165, 56)-Net over F2 — Digital
Digital (80, 165, 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
(165−85, 165, 170)-Net over F2 — Upper bound on s (digital)
There is no digital (80, 165, 171)-net over F2, because
- 5 times m-reduction [i] would yield digital (80, 160, 171)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2160, 171, F2, 80) (dual of [171, 11, 81]-code), but
- residual code [i] would yield linear OA(280, 90, F2, 40) (dual of [90, 10, 41]-code), but
- residual code [i] would yield linear OA(240, 49, F2, 20) (dual of [49, 9, 21]-code), but
- residual code [i] would yield linear OA(280, 90, F2, 40) (dual of [90, 10, 41]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(2160, 171, F2, 80) (dual of [171, 11, 81]-code), but
(165−85, 165, 191)-Net in Base 2 — Upper bound on s
There is no (80, 165, 192)-net in base 2, because
- 1 times m-reduction [i] would yield (80, 164, 192)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 26 990097 305439 081249 137755 522680 366028 772618 603871 > 2164 [i]