MinT - Best Known (124−65, 124, s)-Nets in Base 5

Best Known (124−65, 124, s)-Nets in Base 5

(124−65, 124, 82)-Net over F₅ — Constructive and digital

Digital (59, 124, 82)-net over F₅, using

t-expansion [i] based on digital (48, 124, 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]

(124−65, 124, 112)-Net over F₅ — Digital

Digital (59, 124, 112)-net over F₅, using

t-expansion [i] based on digital (57, 124, 112)-net over F₅, using
- net from sequence [i] based on digital (57, 111)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 57 and N(F) ≥ 112, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(124−65, 124, 1530)-Net in Base 5 — Upper bound on s

There is no (59, 124, 1531)-net in base 5, because

1 times m-reduction [i] would yield (59, 123, 1531)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 94 519844 983176 680175 747729 641640 614791 126931 975224 004152 715848 199205 348295 537263 025025 > 5¹²³ [i]