MinT - Best Known (127−65, 127, s)-Nets in Base 5

Best Known (127−65, 127, s)-Nets in Base 5

(127−65, 127, 82)-Net over F₅ — Constructive and digital

Digital (62, 127, 82)-net over F₅, using

t-expansion [i] based on digital (48, 127, 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]

(127−65, 127, 120)-Net over F₅ — Digital

Digital (62, 127, 120)-net over F₅, using

t-expansion [i] based on digital (61, 127, 120)-net over F₅, using
- net from sequence [i] based on digital (61, 119)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 61 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(127−65, 127, 1783)-Net in Base 5 — Upper bound on s

There is no (62, 127, 1784)-net in base 5, because

1 times m-reduction [i] would yield (62, 126, 1784)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 11787 411365 304302 584019 577543 589354 023652 446714 513195 152289 396270 448237 732127 486097 682433 > 5¹²⁶ [i]