MinT - Best Known (106, 106+37, s)-Nets in Base 5

Best Known (106, 106+37, s)-Nets in Base 5

(106, 106+37, 410)-Net over F₅ — Constructive and digital

Digital (106, 143, 410)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 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 (87, 124, 400)-net over F₅, using
  - trace code for nets [i] based on digital (25, 62, 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]

(106, 106+37, 2152)-Net over F₅ — Digital

Digital (106, 143, 2152)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(106, 106+37, 616818)-Net in Base 5 — Upper bound on s

There is no (106, 143, 616819)-net in base 5, because

1 times m-reduction [i] would yield (106, 142, 616819)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1793 671672 897373 122065 679290 792201 379689 481418 639467 621018 689821 282019 843313 547251 299481 681382 931001 > 5¹⁴² [i]