MinT - Best Known (67−49, 67, s)-Nets in Base 4

Best Known (67−49, 67, s)-Nets in Base 4

(67−49, 67, 33)-Net over F₄ — Constructive and digital

Digital (18, 67, 33)-net over F₄, using

t-expansion [i] based on digital (15, 67, 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]

(67−49, 67, 41)-Net over F₄ — Digital

Digital (18, 67, 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]

(67−49, 67, 85)-Net in Base 4 — Upper bound on s

There is no (18, 67, 86)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁶⁷, 86, S₄, 49), but
- the linear programming bound shows that MÂ â‰¥ 340980 973122 113947 503586 204711 540242 664152 004263 149568 / 13 752615 367645 > 4⁶⁷ [i]