MinT - Best Known (127−83, 127, s)-Nets in Base 5

Best Known (127−83, 127, s)-Nets in Base 5

(127−83, 127, 78)-Net over F₅ — Constructive and digital

Digital (44, 127, 78)-net over F₅, using

t-expansion [i] based on digital (38, 127, 78)-net over F₅, using
- net from sequence [i] based on digital (38, 77)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 38 and N(F) ≥ 78, using
    - an explicitly constructive algebraic function field [i]

(127−83, 127, 84)-Net over F₅ — Digital

Digital (44, 127, 84)-net over F₅, using

t-expansion [i] based on digital (43, 127, 84)-net over F₅, using
- net from sequence [i] based on digital (43, 83)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 43 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(127−83, 127, 537)-Net in Base 5 — Upper bound on s

There is no (44, 127, 538)-net in base 5, because

1 times m-reduction [i] would yield (44, 126, 538)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 11983 792219 055151 124851 350920 636333 805333 300760 745763 160378 662681 359736 790398 960704 869225 > 5¹²⁶ [i]