MinT - Best Known (52−35, 52, s)-Nets in Base 4

Best Known (52−35, 52, s)-Nets in Base 4

(52−35, 52, 33)-Net over F₄ — Constructive and digital

Digital (17, 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]

(52−35, 52, 40)-Net over F₄ — Digital

Digital (17, 52, 40)-net over F₄, using

net from sequence [i] based on digital (17, 39)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 17 and N(F) ≥ 40, using
  - Drinfeld modules of rank 1 [i]

(52−35, 52, 137)-Net in Base 4 — Upper bound on s

There is no (17, 52, 138)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵², 138, S₄, 35), but
- the linear programming bound shows that MÂ â‰¥ 14 564003 487067 138043 331719 274369 344824 206731 992337 689402 817088 211676 748054 528000 / 703008 083798 117523 214095 285089 936359 793876 700683 > 4⁵² [i]