MinT - Best Known (65, 65+77, s)-Nets in Base 5

Best Known (65, 65+77, s)-Nets in Base 5

(65, 65+77, 82)-Net over F₅ — Constructive and digital

Digital (65, 142, 82)-net over F₅, using

t-expansion [i] based on digital (48, 142, 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]

(65, 65+77, 120)-Net over F₅ — Digital

Digital (65, 142, 120)-net over F₅, using

t-expansion [i] based on digital (61, 142, 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]

(65, 65+77, 1445)-Net in Base 5 — Upper bound on s

There is no (65, 142, 1446)-net in base 5, because

1 times m-reduction [i] would yield (65, 141, 1446)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 363 544687 709313 234725 378515 401763 648309 695758 160422 805801 192759 545238 639967 386856 620181 885175 488065 > 5¹⁴¹ [i]