MinT - Best Known (71, 71+71, s)-Nets in Base 5

Best Known (71, 71+71, s)-Nets in Base 5

(71, 71+71, 92)-Net over F₅ — Constructive and digital

Digital (71, 142, 92)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 40, 20)-net over F₅, using
  - net from sequence [i] based on digital (5, 19)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 5 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 102, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(71, 71+71, 139)-Net over F₅ — Digital

Digital (71, 142, 139)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(71, 71+71, 2249)-Net in Base 5 — Upper bound on s

There is no (71, 142, 2250)-net in base 5, because

1 times m-reduction [i] would yield (71, 141, 2250)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 360 192621 602109 407048 789204 910492 386338 634096 238131 687546 506552 452470 028868 983793 577001 133854 684361 > 5¹⁴¹ [i]