MinT - Best Known (63−43, 63, s)-Nets in Base 4

Best Known (63−43, 63, s)-Nets in Base 4

(63−43, 63, 33)-Net over F₄ — Constructive and digital

Digital (20, 63, 33)-net over F₄, using

t-expansion [i] based on digital (15, 63, 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]

(63−43, 63, 41)-Net over F₄ — Digital

Digital (20, 63, 41)-net over F₄, using

t-expansion [i] based on digital (18, 63, 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]

(63−43, 63, 144)-Net in Base 4 — Upper bound on s

There is no (20, 63, 145)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁶³, 145, S₄, 43), but
- the linear programming bound shows that MÂ â‰¥ 410852 595144 601422 661736 194475 352548 234222 916103 520325 789662 026727 829058 017732 594422 665880 071589 138518 417655 328617 787562 131456 / 4712 214848 200787 196384 454868 078661 057064 361851 730871 257084 069591 637575 047639 691128 659913 > 4⁶³ [i]