MinT - Best Known (115−59, 115, s)-Nets in Base 5

Best Known (115−59, 115, s)-Nets in Base 5

(115−59, 115, 82)-Net over F₅ — Constructive and digital

Digital (56, 115, 82)-net over F₅, using

t-expansion [i] based on digital (48, 115, 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]

(115−59, 115, 108)-Net over F₅ — Digital

Digital (56, 115, 108)-net over F₅, using

t-expansion [i] based on digital (55, 115, 108)-net over F₅, using
- net from sequence [i] based on digital (55, 107)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 55 and N(F) ≥ 108, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(115−59, 115, 1610)-Net in Base 5 — Upper bound on s

There is no (56, 115, 1611)-net in base 5, because

1 times m-reduction [i] would yield (56, 114, 1611)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 48 222994 894821 353674 923871 736774 338653 146177 448778 123859 255617 735380 913894 967485 > 5¹¹⁴ [i]