MinT - Best Known (111, 111+34, s)-Nets in Base 5

Best Known (111, 111+34, s)-Nets in Base 5

(111, 111+34, 504)-Net over F₅ — Constructive and digital

Digital (111, 145, 504)-net over F₅, using

5¹ times duplication [i] based on digital (110, 144, 504)-net over F₅, using
- t-expansion [i] based on digital (109, 144, 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 (55, 90, 252)-net over F₅, using
      - trace code for nets [i] based on digital (10, 45, 126)-net over F₂₅, using
        net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(111, 111+34, 3894)-Net over F₅ — Digital

Digital (111, 145, 3894)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 111+34, 1643179)-Net in Base 5 — Upper bound on s

There is no (111, 145, 1643180)-net in base 5, because

the generalized Rao bound for nets shows that 5^mÂ â‰¥ 224209 371993 663906 109230 070880 809691 818536 857543 967322 523696 685440 773468 108930 181643 547691 801047 369905 > 5¹⁴⁵ [i]