MinT - Best Known (65, 65+191, s)-Nets in Base 2

Best Known (65, 65+191, s)-Nets in Base 2

(65, 65+191, 43)-Net over F₂ — Constructive and digital

Digital (65, 256, 43)-net over F₂, using

t-expansion [i] based on digital (59, 256, 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]

(65, 65+191, 48)-Net over F₂ — Digital

Digital (65, 256, 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]

(65, 65+191, 82)-Net in Base 2 — Upper bound on s

There is no (65, 256, 83)-net in base 2, because

14 times m-reduction [i] would yield (65, 242, 83)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁴², 83, S₂, 3, 177), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 735 008378 947807 881106 651760 309054 102584 233301 100257 486578 588179 737729 826816 / 89 > 2²⁴² [i]