MinT - Best Known (123−45, 123, s)-Nets in Base 5

Best Known (123−45, 123, s)-Nets in Base 5

(123−45, 123, 252)-Net over F₅ — Constructive and digital

Digital (78, 123, 252)-net over F₅, using

13 times m-reduction [i] based on digital (78, 136, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 68, 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]

(123−45, 123, 381)-Net over F₅ — Digital

Digital (78, 123, 381)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−45, 123, 17001)-Net in Base 5 — Upper bound on s

There is no (78, 123, 17002)-net in base 5, because

1 times m-reduction [i] would yield (78, 122, 17002)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 18 816445 971204 826229 913715 428046 881984 132615 058740 168413 124637 832072 835346 245635 562625 > 5¹²² [i]