MinT - Best Known (139−31, 139, s)-Nets in Base 5

Best Known (139−31, 139, s)-Nets in Base 5

(139−31, 139, 504)-Net over F₅ — Constructive and digital

Digital (108, 139, 504)-net over F₅, using

3 times m-reduction [i] based on digital (108, 142, 504)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (37, 54, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 27, 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]
  2. digital (54, 88, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 44, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(139−31, 139, 5230)-Net over F₅ — Digital

Digital (108, 139, 5230)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−31, 139, 4327425)-Net in Base 5 — Upper bound on s

There is no (108, 139, 4327426)-net in base 5, because

1 times m-reduction [i] would yield (108, 138, 4327426)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 869863 182250 999243 988481 101392 428367 761344 844325 406377 803035 738717 775042 947283 099984 343920 139305 > 5¹³⁸ [i]