MinT - Best Known (54, 54+91, s)-Nets in Base 5

Best Known (54, 54+91, s)-Nets in Base 5

(54, 54+91, 82)-Net over F₅ — Constructive and digital

Digital (54, 145, 82)-net over F₅, using

t-expansion [i] based on digital (48, 145, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(54, 54+91, 104)-Net over F₅ — Digital

Digital (54, 145, 104)-net over F₅, using

t-expansion [i] based on digital (51, 145, 104)-net over F₅, using
- net from sequence [i] based on digital (51, 103)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 51 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(54, 54+91, 727)-Net in Base 5 — Upper bound on s

There is no (54, 145, 728)-net in base 5, because

1 times m-reduction [i] would yield (54, 144, 728)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 46839 191501 894765 748195 861102 517007 149581 089699 945091 932440 072267 423168 228012 636062 596061 996169 632865 > 5¹⁴⁴ [i]