MinT - Best Known (251−135, 251, s)-Nets in Base 2

Best Known (251−135, 251, s)-Nets in Base 2

(251−135, 251, 57)-Net over F₂ — Constructive and digital

Digital (116, 251, 57)-net over F₂, using

t-expansion [i] based on digital (110, 251, 57)-net over F₂, using
- net from sequence [i] based on digital (110, 56)-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 8 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]

(251−135, 251, 73)-Net over F₂ — Digital

Digital (116, 251, 73)-net over F₂, using

t-expansion [i] based on digital (114, 251, 73)-net over F₂, using
- net from sequence [i] based on digital (114, 72)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 114 and N(F) ≥ 73, using
    - Drinfeld modules of rank 1 [i]

(251−135, 251, 244)-Net in Base 2 — Upper bound on s

There is no (116, 251, 245)-net in base 2, because

27 times m-reduction [i] would yield (116, 224, 245)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²⁴, 245, S₂, 108), but
  - the linear programming bound shows that MÂ â‰¥ 227 481559 590216 982433 849553 220650 758912 788551 722735 940858 156798 416424 796160 / 8 133191 > 2²²⁴ [i]