MinT - Best Known (64, 64+61, s)-Nets in Base 5

Best Known (64, 64+61, s)-Nets in Base 5

(64, 64+61, 88)-Net over F₅ — Constructive and digital

Digital (64, 125, 88)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 33, 16)-net over F₅, using
  - net from sequence [i] based on digital (3, 15)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 92, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(64, 64+61, 136)-Net over F₅ — Digital

Digital (64, 125, 136)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(64, 64+61, 2310)-Net in Base 5 — Upper bound on s

There is no (64, 125, 2311)-net in base 5, because

1 times m-reduction [i] would yield (64, 124, 2311)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 473 335068 156084 104012 489826 907445 207930 192177 798663 440918 823382 912157 974551 786037 985929 > 5¹²⁴ [i]