MinT - Best Known (129−83, 129, s)-Nets in Base 5

Best Known (129−83, 129, s)-Nets in Base 5

(129−83, 129, 78)-Net over F₅ — Constructive and digital

Digital (46, 129, 78)-net over F₅, using

t-expansion [i] based on digital (38, 129, 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]

(129−83, 129, 88)-Net over F₅ — Digital

Digital (46, 129, 88)-net over F₅, using

t-expansion [i] based on digital (45, 129, 88)-net over F₅, using
- net from sequence [i] based on digital (45, 87)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 45 and N(F) ≥ 88, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(129−83, 129, 584)-Net in Base 5 — Upper bound on s

There is no (46, 129, 585)-net in base 5, because

1 times m-reduction [i] would yield (46, 128, 585)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 313434 881056 765017 036936 922415 366899 364815 019862 590140 828483 719371 023406 347487 608765 911045 > 5¹²⁸ [i]