MinT - Best Known (84, 84+157, s)-Nets in Base 2

Best Known (84, 84+157, s)-Nets in Base 2

(84, 84+157, 51)-Net over F₂ — Constructive and digital

Digital (84, 241, 51)-net over F₂, using

t-expansion [i] based on digital (80, 241, 51)-net over F₂, using
- net from sequence [i] based on digital (80, 50)-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 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]

(84, 84+157, 57)-Net over F₂ — Digital

Digital (84, 241, 57)-net over F₂, using

t-expansion [i] based on digital (83, 241, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

(84, 84+157, 121)-Net in Base 2 — Upper bound on s

There is no (84, 241, 122)-net in base 2, because

4 times m-reduction [i] would yield (84, 237, 122)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²³⁷, 122, S₂, 2, 153), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 21 202164 777340 611954 999570 008915 022189 929806 762507 427497 459274 415511 437312 / 77 > 2²³⁷ [i]