MinT - Best Known (127−63, 127, s)-Nets in Base 5

Best Known (127−63, 127, s)-Nets in Base 5

(127−63, 127, 84)-Net over F₅ — Constructive and digital

Digital (64, 127, 84)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 33, 12)-net over F₅, using
  - net from sequence [i] based on digital (2, 11)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 2 and N(F) ≥ 12, 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]

(127−63, 127, 130)-Net over F₅ — Digital

Digital (64, 127, 130)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−63, 127, 2129)-Net in Base 5 — Upper bound on s

There is no (64, 127, 2130)-net in base 5, because

1 times m-reduction [i] would yield (64, 126, 2130)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 11778 878258 959003 340858 502952 566800 846702 308956 229882 460764 628199 406342 937656 767882 366825 > 5¹²⁶ [i]