MinT - Best Known (68−49, 68, s)-Nets in Base 4

Best Known (68−49, 68, s)-Nets in Base 4

(68−49, 68, 33)-Net over F₄ — Constructive and digital

Digital (19, 68, 33)-net over F₄, using

t-expansion [i] based on digital (15, 68, 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]

(68−49, 68, 41)-Net over F₄ — Digital

Digital (19, 68, 41)-net over F₄, using

t-expansion [i] based on digital (18, 68, 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]

(68−49, 68, 98)-Net in Base 4 — Upper bound on s

There is no (19, 68, 99)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁶⁸, 99, S₄, 49), but
- the linear programming bound shows that MÂ â‰¥ 69 793713 170008 775407 317380 828242 293910 072404 471707 443783 127810 441216 / 673 220107 091652 990724 765625 > 4⁶⁸ [i]