MinT - Best Known (251−162, 251, s)-Nets in Base 2

Best Known (251−162, 251, s)-Nets in Base 2

(251−162, 251, 52)-Net over F₂ — Constructive and digital

Digital (89, 251, 52)-net over F₂, using

t-expansion [i] based on digital (85, 251, 52)-net over F₂, using
- net from sequence [i] based on digital (85, 51)-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 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]

(251−162, 251, 57)-Net over F₂ — Digital

Digital (89, 251, 57)-net over F₂, using

t-expansion [i] based on digital (83, 251, 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]

(251−162, 251, 128)-Net in Base 2 — Upper bound on s

There is no (89, 251, 129)-net in base 2, because

1 times m-reduction [i] would yield (89, 250, 129)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁵⁰, 129, S₂, 2, 161), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 57896 044618 658097 711785 492504 343953 926634 992332 820282 019728 792003 956564 819968 / 27 > 2²⁵⁰ [i]