MinT - Best Known (57−45, 57, s)-Nets in Base 8

Best Known (57−45, 57, s)-Nets in Base 8

(57−45, 57, 48)-Net over F₈ — Constructive and digital

Digital (12, 57, 48)-net over F₈, using

t-expansion [i] based on digital (11, 57, 48)-net over F₈, using
- net from sequence [i] based on digital (11, 47)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
    - a function field by SÃ©mirat [i]

(57−45, 57, 49)-Net over F₈ — Digital

Digital (12, 57, 49)-net over F₈, using

net from sequence [i] based on digital (12, 48)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 12 and N(F) ≥ 49, using
  - a curve from the manYPoints database [i]

(57−45, 57, 232)-Net in Base 8 — Upper bound on s

There is no (12, 57, 233)-net in base 8, because

extracting embedded orthogonal array [i] would yield OA(8⁵⁷, 233, S₈, 45), but
- the linear programming bound shows that MÂ â‰¥ 237931 612421 660871 264709 152821 634197 832300 583413 225860 593396 351519 378998 735085 682686 164364 049929 997846 026708 615535 001600 / 75 581854 371945 882889 275980 307190 955914 447030 417585 005366 757576 059679 > 8⁵⁷ [i]