MinT - Best Known (137−57, 137, s)-Nets in Base 5

Best Known (137−57, 137, s)-Nets in Base 5

(137−57, 137, 252)-Net over F₅ — Constructive and digital

Digital (80, 137, 252)-net over F₅, using

3 times m-reduction [i] based on digital (80, 140, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 70, 126)-net over F₂₅, using
  - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
      - the Hermitian function field over F₂₅ [i]

(137−57, 137, 257)-Net over F₅ — Digital

Digital (80, 137, 257)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−57, 137, 6993)-Net in Base 5 — Upper bound on s

There is no (80, 137, 6994)-net in base 5, because

1 times m-reduction [i] would yield (80, 136, 6994)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 115188 700665 250726 005660 555904 899627 924620 044678 139367 323008 638273 063515 156682 523274 332593 829569 > 5¹³⁶ [i]