MinT - Best Known (52−36, 52, s)-Nets in Base 4

Best Known (52−36, 52, s)-Nets in Base 4

(52−36, 52, 33)-Net over F₄ — Constructive and digital

Digital (16, 52, 33)-net over F₄, using

t-expansion [i] based on digital (15, 52, 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]

(52−36, 52, 36)-Net over F₄ — Digital

Digital (16, 52, 36)-net over F₄, using

net from sequence [i] based on digital (16, 35)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 16 and N(F) ≥ 36, using
  - Drinfeld modules of rank 1 [i]

(52−36, 52, 118)-Net in Base 4 — Upper bound on s

There is no (16, 52, 119)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵², 119, S₄, 36), but
- the linear programming bound shows that MÂ â‰¥ 357 859286 624924 007499 700605 808588 465393 308876 978703 840422 989308 837582 985335 138623 209171 597503 690371 497984 / 17 092119 397133 148679 798998 224288 670000 327306 475640 461791 263157 951334 577147 > 4⁵² [i]