MinT - Best Known (252−174, 252, s)-Nets in Base 2

Best Known (252−174, 252, s)-Nets in Base 2

(252−174, 252, 50)-Net over F₂ — Constructive and digital

Digital (78, 252, 50)-net over F₂, using

t-expansion [i] based on digital (75, 252, 50)-net over F₂, using
- net from sequence [i] based on digital (75, 49)-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 1 place 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]

(252−174, 252, 52)-Net over F₂ — Digital

Digital (78, 252, 52)-net over F₂, using

t-expansion [i] based on digital (77, 252, 52)-net over F₂, using
- net from sequence [i] based on digital (77, 51)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 77 and N(F) ≥ 52, using
    - Drinfeld modules of rank 1 [i]

(252−174, 252, 113)-Net in Base 2 — Upper bound on s

There is no (78, 252, 114)-net in base 2, because

31 times m-reduction [i] would yield (78, 221, 114)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²²¹, 114, S₂, 2, 143), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 13 479973 333575 319897 333507 543509 815336 818572 211270 286240 551805 124608 / 3 > 2²²¹ [i]