MinT - Best Known (109, 109+29, s)-Nets in Base 5

Best Known (109, 109+29, s)-Nets in Base 5

(109, 109+29, 520)-Net over F₅ — Constructive and digital

Digital (109, 138, 520)-net over F₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 12, 16)-net over F₅, using
  - net from sequence [i] based on digital (3, 15)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 3 and N(F) ≥ 16, using
      - an explicitly constructive algebraic function field [i]
2. digital (34, 48, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 24, 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]
3. digital (49, 78, 252)-net over F₅, using
  - trace code for nets [i] based on digital (10, 39, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(109, 109+29, 7882)-Net over F₅ — Digital

Digital (109, 138, 7882)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(109, 109+29, large)-Net in Base 5 — Upper bound on s

There is no (109, 138, large)-net in base 5, because

27 times m-reduction [i] would yield (109, 111, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]