MinT - Best Known (57−39, 57, s)-Nets in Base 4

Best Known (57−39, 57, s)-Nets in Base 4

(57−39, 57, 33)-Net over F₄ — Constructive and digital

Digital (18, 57, 33)-net over F₄, using

t-expansion [i] based on digital (15, 57, 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]

(57−39, 57, 41)-Net over F₄ — Digital

Digital (18, 57, 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]

(57−39, 57, 133)-Net in Base 4 — Upper bound on s

There is no (18, 57, 134)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁷, 134, S₄, 39), but
- the linear programming bound shows that MÂ â‰¥ 2118 872152 947773 735205 614860 840104 993723 173522 547123 836257 668222 520389 144284 698869 497944 229365 678080 000000 / 96823 132929 101376 060431 157559 643789 049324 330608 249156 649224 680559 609039 > 4⁵⁷ [i]