MinT - Best Known (226−115, 226, s)-Nets in Base 2

Best Known (226−115, 226, s)-Nets in Base 2

(226−115, 226, 57)-Net over F₂ — Constructive and digital

Digital (111, 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]

(226−115, 226, 72)-Net over F₂ — Digital

Digital (111, 226, 72)-net over F₂, using

t-expansion [i] based on digital (110, 226, 72)-net over F₂, using
- net from sequence [i] based on digital (110, 71)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 110 and N(F) ≥ 72, using
    - Drinfeld modules of rank 1 [i]

(226−115, 226, 232)-Net over F₂ — Upper bound on s (digital)

There is no digital (111, 226, 233)-net over F₂, because

3 times m-reduction [i] would yield digital (111, 223, 233)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2²²³, 233, F₂, 112) (dual of [233, 10, 113]-code), but
  - residual code [i] would yield linear OA(2¹¹¹, 120, F₂, 56) (dual of [120, 9, 57]-code), but
    - residual code [i] would yield linear OA(2⁵⁵, 63, F₂, 28) (dual of [63, 8, 29]-code), but
      - â€œBJVâ€ bound on codes from Brouwerâ€™s database [i]

(226−115, 226, 234)-Net in Base 2 — Upper bound on s

There is no (111, 226, 235)-net in base 2, because

11 times m-reduction [i] would yield (111, 215, 235)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2²¹⁵, 235, S₂, 104), but
  - the linear programming bound shows that MÂ â‰¥ 8195 823786 813794 497578 772586 453967 724785 691904 452334 034255 497515 761664 / 117183 > 2²¹⁵ [i]