MinT - Best Known (138−51, 138, s)-Nets in Base 5

Best Known (138−51, 138, s)-Nets in Base 5

(138−51, 138, 252)-Net over F₅ — Constructive and digital

Digital (87, 138, 252)-net over F₅, using

t-expansion [i] based on digital (85, 138, 252)-net over F₅, using
- 12 times m-reduction [i] based on digital (85, 150, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 75, 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]

(138−51, 138, 402)-Net over F₅ — Digital

Digital (87, 138, 402)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−51, 138, 17197)-Net in Base 5 — Upper bound on s

There is no (87, 138, 17198)-net in base 5, because

1 times m-reduction [i] would yield (87, 137, 17198)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 574531 351288 690191 389298 995275 733393 272969 905000 739797 945082 820563 740536 751263 872329 387378 877113 > 5¹³⁷ [i]