MinT - Best Known (128−63, 128, s)-Nets in Base 5

Best Known (128−63, 128, s)-Nets in Base 5

(128−63, 128, 88)-Net over F₅ — Constructive and digital

Digital (65, 128, 88)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (3, 34, 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, 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]

(128−63, 128, 134)-Net over F₅ — Digital

Digital (65, 128, 134)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−63, 128, 2244)-Net in Base 5 — Upper bound on s

There is no (65, 128, 2245)-net in base 5, because

1 times m-reduction [i] would yield (65, 127, 2245)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 59129 697651 157940 034538 677930 421486 629156 601388 398382 138647 438676 025527 912075 603457 543837 > 5¹²⁷ [i]