MinT - Best Known (51−35, 51, s)-Nets in Base 4

Best Known (51−35, 51, s)-Nets in Base 4

(51−35, 51, 33)-Net over F₄ — Constructive and digital

Digital (16, 51, 33)-net over F₄, using

t-expansion [i] based on digital (15, 51, 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]

(51−35, 51, 36)-Net over F₄ — Digital

Digital (16, 51, 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]

(51−35, 51, 123)-Net in Base 4 — Upper bound on s

There is no (16, 51, 124)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵¹, 124, S₄, 35), but
- the linear programming bound shows that MÂ â‰¥ 691 446239 734930 013936 659131 075507 527072 529250 608218 542950 198517 596797 334894 673920 / 123 717113 118431 928293 789080 421222 899797 731711 329751 > 4⁵¹ [i]