MinT - Best Known (77−57, 77, s)-Nets in Base 4

Best Known (77−57, 77, s)-Nets in Base 4

(77−57, 77, 33)-Net over F₄ — Constructive and digital

Digital (20, 77, 33)-net over F₄, using

t-expansion [i] based on digital (15, 77, 33)-net over F₄, using
- net from sequence [i] based on digital (15, 32)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(77−57, 77, 41)-Net over F₄ — Digital

Digital (20, 77, 41)-net over F₄, using

t-expansion [i] based on digital (18, 77, 41)-net over F₄, using
- net from sequence [i] based on digital (18, 40)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 18 and N(F) ≥ 41, using
    - Drinfeld modules of rank 1 [i]

(77−57, 77, 89)-Net in Base 4 — Upper bound on s

There is no (20, 77, 90)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁷⁷, 90, S₄, 57), but
- the linear programming bound shows that MÂ â‰¥ 812 594910 355982 022521 248766 990253 358928 698493 894656 / 29029 > 4⁷⁷ [i]