MinT - Best Known (72−52, 72, s)-Nets in Base 4

Best Known (72−52, 72, s)-Nets in Base 4

(72−52, 72, 33)-Net over F₄ — Constructive and digital

Digital (20, 72, 33)-net over F₄, using

t-expansion [i] based on digital (15, 72, 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]

(72−52, 72, 41)-Net over F₄ — Digital

Digital (20, 72, 41)-net over F₄, using

t-expansion [i] based on digital (18, 72, 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]

(72−52, 72, 98)-Net in Base 4 — Upper bound on s

There is no (20, 72, 99)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁷², 99, S₄, 52), but
- the linear programming bound shows that MÂ â‰¥ 8994 332944 549956 910563 258210 239995 230668 649109 461390 448251 633664 / 316 939821 962551 080475 > 4⁷² [i]