MinT - Best Known (131−63, 131, s)-Nets in Base 5

Best Known (131−63, 131, s)-Nets in Base 5

(131−63, 131, 93)-Net over F₅ — Constructive and digital

Digital (68, 131, 93)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (6, 37, 21)-net over F₅, using
  - net from sequence [i] based on digital (6, 20)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 6 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 94, 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]

(131−63, 131, 148)-Net over F₅ — Digital

Digital (68, 131, 148)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(131−63, 131, 2626)-Net in Base 5 — Upper bound on s

There is no (68, 131, 2627)-net in base 5, because

1 times m-reduction [i] would yield (68, 130, 2627)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 7 376257 166153 415918 421443 349053 075563 052976 883396 866560 379112 844603 532637 478587 997441 885525 > 5¹³⁰ [i]