MinT - Best Known (53, 53+89, s)-Nets in Base 5

Best Known (53, 53+89, s)-Nets in Base 5

(53, 53+89, 82)-Net over F₅ — Constructive and digital

Digital (53, 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]

(53, 53+89, 104)-Net over F₅ — Digital

Digital (53, 142, 104)-net over F₅, using

t-expansion [i] based on digital (51, 142, 104)-net over F₅, using
- net from sequence [i] based on digital (51, 103)-sequence over F₅, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 51 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₅ by Niederreiter/Xing [i]

(53, 53+89, 717)-Net in Base 5 — Upper bound on s

There is no (53, 142, 718)-net in base 5, because

1 times m-reduction [i] would yield (53, 141, 718)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 370 649897 191876 006287 198371 728734 430321 653452 238506 940059 118808 170620 432977 209015 832190 970990 386625 > 5¹⁴¹ [i]