MinT - Best Known (109−55, 109, s)-Nets in Base 5

Best Known (109−55, 109, s)-Nets in Base 5

(109−55, 109, 82)-Net over F₅ — Constructive and digital

Digital (54, 109, 82)-net over F₅, using

t-expansion [i] based on digital (48, 109, 82)-net over F₅, using
- net from sequence [i] based on digital (48, 81)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 48 and N(F) ≥ 82, using
    - an explicitly constructive algebraic function field [i]

(109−55, 109, 108)-Net over F₅ — Digital

Digital (54, 109, 108)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−55, 109, 1687)-Net in Base 5 — Upper bound on s

There is no (54, 109, 1688)-net in base 5, because

1 times m-reduction [i] would yield (54, 108, 1688)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 3115 538258 209263 081285 912695 885528 119450 326013 619727 971940 046308 605434 949473 > 5¹⁰⁸ [i]