MinT - Best Known (17−9, 17, s)-Nets in Base 5

Best Known (17−9, 17, s)-Nets in Base 5

(17−9, 17, 28)-Net over F₅ — Constructive and digital

Digital (8, 17, 28)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 5, 12)-net over F₅, using
  - linear code embedding found in a computer search [i]
2. digital (3, 12, 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]

(17−9, 17, 30)-Net over F₅ — Digital

Digital (8, 17, 30)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(17−9, 17, 343)-Net in Base 5 — Upper bound on s

There is no (8, 17, 344)-net in base 5, because

1 times m-reduction [i] would yield (8, 16, 344)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 153746 622465 > 5¹⁶ [i]