MinT - Best Known (168−116, 168, s)-Nets in Base 2

Best Known (168−116, 168, s)-Nets in Base 2

(168−116, 168, 36)-Net over F₂ — Constructive and digital

Digital (52, 168, 36)-net over F₂, using

t-expansion [i] based on digital (51, 168, 36)-net over F₂, using
- net from sequence [i] based on digital (51, 35)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(168−116, 168, 40)-Net over F₂ — Digital

Digital (52, 168, 40)-net over F₂, using

t-expansion [i] based on digital (50, 168, 40)-net over F₂, using
- net from sequence [i] based on digital (50, 39)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 50 and N(F) ≥ 40, using
    - Drinfeld modules of rank 1 [i]

(168−116, 168, 78)-Net in Base 2 — Upper bound on s

There is no (52, 168, 79)-net in base 2, because

17 times m-reduction [i] would yield (52, 151, 79)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁵¹, 79, S₂, 2, 99), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 22835 963083 295358 096932 575511 191922 182123 945984 / 5 > 2¹⁵¹ [i]