MinT - Best Known (168−93, 168, s)-Nets in Base 2

Best Known (168−93, 168, s)-Nets in Base 2

(168−93, 168, 50)-Net over F₂ — Constructive and digital

Digital (75, 168, 50)-net over F₂, using

net from sequence [i] based on digital (75, 49)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 1 place 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]

(168−93, 168, 160)-Net over F₂ — Upper bound on s (digital)

There is no digital (75, 168, 161)-net over F₂, because

13 times m-reduction [i] would yield digital (75, 155, 161)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁵⁵, 161, F₂, 80) (dual of [161, 6, 81]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(2¹⁵⁶, 162, F₂, 80) (dual of [162, 6, 81]-code), but
    - Griesmer bound [i]

(168−93, 168, 162)-Net in Base 2 — Upper bound on s

There is no (75, 168, 163)-net in base 2, because

1 times m-reduction [i] would yield (75, 167, 163)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 200 855268 977588 036038 190318 675297 468133 289507 555168 > 2¹⁶⁷ [i]