MinT - Best Known (51, 51+91, s)-Nets in Base 5

Best Known (51, 51+91, s)-Nets in Base 5

(51, 51+91, 82)-Net over F₅ — Constructive and digital

Digital (51, 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]

(51, 51+91, 104)-Net over F₅ — Digital

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]

(51, 51+91, 649)-Net in Base 5 — Upper bound on s

There is no (51, 142, 650)-net in base 5, because

1 times m-reduction [i] would yield (51, 141, 650)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 358 806119 793291 624870 889825 544502 239527 354855 407147 820771 983040 771642 082199 042770 259691 766014 781225 > 5¹⁴¹ [i]