MinT - Best Known (249−163, 249, s)-Nets in Base 2

Best Known (249−163, 249, s)-Nets in Base 2

(249−163, 249, 52)-Net over F₂ — Constructive and digital

Digital (86, 249, 52)-net over F₂, using

t-expansion [i] based on digital (85, 249, 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]

(249−163, 249, 57)-Net over F₂ — Digital

Digital (86, 249, 57)-net over F₂, using

t-expansion [i] based on digital (83, 249, 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]

(249−163, 249, 124)-Net in Base 2 — Upper bound on s

There is no (86, 249, 125)-net in base 2, because

7 times m-reduction [i] would yield (86, 242, 125)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁴², 125, S₂, 2, 156), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1356 938545 749799 165119 972480 570561 420155 507632 800475 359837 393562 592731 987968 / 157 > 2²⁴² [i]