MinT - Best Known (103, 138, s)-Nets in Base 5

Best Known (103, 138, s)-Nets in Base 5

(103, 138, 410)-Net over F₅ — Constructive and digital

Digital (103, 138, 410)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 18, 10)-net over F₅, using
  - net from sequence [i] based on digital (1, 9)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 1 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (85, 120, 400)-net over F₅, using
  - trace code for nets [i] based on digital (25, 60, 200)-net over F₂₅, using
    - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
        a function field by GarcÃa and Quoos [i]

(103, 138, 2342)-Net over F₅ — Digital

Digital (103, 138, 2342)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(103, 138, 770467)-Net in Base 5 — Upper bound on s

There is no (103, 138, 770468)-net in base 5, because

1 times m-reduction [i] would yield (103, 137, 770468)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 573980 343480 689032 950403 015919 376171 025141 487073 793180 449846 305006 529365 799375 232465 000181 151889 > 5¹³⁷ [i]