MinT - Best Known (257−167, 257, s)-Nets in Base 2

Best Known (257−167, 257, s)-Nets in Base 2

(257−167, 257, 53)-Net over F₂ — Constructive and digital

Digital (90, 257, 53)-net over F₂, using

net from sequence [i] based on digital (90, 52)-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 4 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]

(257−167, 257, 57)-Net over F₂ — Digital

Digital (90, 257, 57)-net over F₂, using

t-expansion [i] based on digital (83, 257, 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]

(257−167, 257, 129)-Net in Base 2 — Upper bound on s

There is no (90, 257, 130)-net in base 2, because

4 times m-reduction [i] would yield (90, 253, 130)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁵³, 130, S₂, 2, 163), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 694752 535423 897172 541425 910052 127447 119619 907993 843384 236745 504047 478777 839616 / 41 > 2²⁵³ [i]