MinT - Best Known (122−59, 122, s)-Nets in Base 5

Best Known (122−59, 122, s)-Nets in Base 5

(122−59, 122, 88)-Net over F₅ — Constructive and digital

Digital (63, 122, 88)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 32, 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, 90, 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]

(122−59, 122, 137)-Net over F₅ — Digital

Digital (63, 122, 137)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(122−59, 122, 2385)-Net in Base 5 — Upper bound on s

There is no (63, 122, 2386)-net in base 5, because

1 times m-reduction [i] would yield (63, 121, 2386)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3 775420 193646 053501 194808 611788 528952 739687 851748 280609 694916 379527 695300 973636 740425 > 5¹²¹ [i]