MinT - Best Known (63, 63+69, s)-Nets in Base 5

Best Known (63, 63+69, s)-Nets in Base 5

(63, 63+69, 82)-Net over F₅ — Constructive and digital

Digital (63, 132, 82)-net over F₅, using

t-expansion [i] based on digital (48, 132, 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]

(63, 63+69, 120)-Net over F₅ — Digital

Digital (63, 132, 120)-net over F₅, using

t-expansion [i] based on digital (61, 132, 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]

(63, 63+69, 1644)-Net in Base 5 — Upper bound on s

There is no (63, 132, 1645)-net in base 5, because

1 times m-reduction [i] would yield (63, 131, 1645)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 37 244379 949146 604565 703171 962015 248362 843971 312121 158038 223779 187280 064308 046254 626053 134745 > 5¹³¹ [i]