MinT - Best Known (251−172, 251, s)-Nets in Base 2

Best Known (251−172, 251, s)-Nets in Base 2

(251−172, 251, 50)-Net over F₂ — Constructive and digital

Digital (79, 251, 50)-net over F₂, using

t-expansion [i] based on digital (75, 251, 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]

(251−172, 251, 52)-Net over F₂ — Digital

Digital (79, 251, 52)-net over F₂, using

t-expansion [i] based on digital (77, 251, 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]

(251−172, 251, 114)-Net in Base 2 — Upper bound on s

There is no (79, 251, 115)-net in base 2, because

28 times m-reduction [i] would yield (79, 223, 115)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²²³, 115, S₂, 2, 144), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 431 359146 674410 236714 672241 392314 090778 194310 760649 159697 657763 987456 / 29 > 2²²³ [i]