MinT - Best Known (226−107, 226, s)-Nets in Base 2

Best Known (226−107, 226, s)-Nets in Base 2

(226−107, 226, 57)-Net over F₂ — Constructive and digital

Digital (119, 226, 57)-net over F₂, using

t-expansion [i] based on digital (110, 226, 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]

(226−107, 226, 73)-Net over F₂ — Digital

Digital (119, 226, 73)-net over F₂, using

t-expansion [i] based on digital (114, 226, 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]

(226−107, 226, 255)-Net in Base 2 — Upper bound on s

There is no (119, 226, 256)-net in base 2, because

1 times m-reduction [i] would yield (119, 225, 256)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²²⁵, 256, S₂, 106), but
  - the linear programming bound shows that MÂ â‰¥ 4 533712 341462 452600 463527 949755 696668 157794 361414 843709 255199 826856 399742 173184 / 59446 678005 > 2²²⁵ [i]