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

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

(65, 65+65, 84)-Net over F₅ — Constructive and digital

Digital (65, 130, 84)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (2, 34, 12)-net over F₅, using
  - net from sequence [i] based on digital (2, 11)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 2 and N(F) ≥ 12, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 96, 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]

(65, 65+65, 129)-Net over F₅ — Digital

Digital (65, 130, 129)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+65, 2078)-Net in Base 5 — Upper bound on s

There is no (65, 130, 2079)-net in base 5, because

1 times m-reduction [i] would yield (65, 129, 2079)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 488271 669276 644460 845464 278800 490410 212163 709438 261846 253774 065007 189252 840670 381290 708865 > 5¹²⁹ [i]