MinT - Best Known (247−181, 247, s)-Nets in Base 2

Best Known (247−181, 247, s)-Nets in Base 2

(247−181, 247, 43)-Net over F₂ — Constructive and digital

Digital (66, 247, 43)-net over F₂, using

t-expansion [i] based on digital (59, 247, 43)-net over F₂, using
- net from sequence [i] based on digital (59, 42)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(247−181, 247, 48)-Net over F₂ — Digital

Digital (66, 247, 48)-net over F₂, using

t-expansion [i] based on digital (65, 247, 48)-net over F₂, using
- net from sequence [i] based on digital (65, 47)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 65 and N(F) ≥ 48, using
    - Drinfeld modules of rank 1 [i]

(247−181, 247, 83)-Net in Base 2 — Upper bound on s

There is no (66, 247, 84)-net in base 2, because

2 times m-reduction [i] would yield (66, 245, 84)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁴⁵, 84, S₂, 3, 179), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 904 625697 166532 776746 648320 380374 280103 671755 200316 906558 262375 061821 325312 / 15 > 2²⁴⁵ [i]