MinT - Best Known (72, 72+25, s)-Nets in Base 5

Best Known (72, 72+25, s)-Nets in Base 5

(72, 72+25, 306)-Net over F₅ — Constructive and digital

Digital (72, 97, 306)-net over F₅, using

5¹ times duplication [i] based on digital (71, 96, 306)-net over F₅, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 26, 54)-net over F₅, using
    - trace code for nets [i] based on digital (1, 13, 27)-net over F₂₅, using
      - net from sequence [i] based on digital (1, 26)-sequence over F₂₅, using
        Niederreiter sequence [i]
  2. digital (45, 70, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 35, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]

(72, 72+25, 1650)-Net over F₅ — Digital

Digital (72, 97, 1650)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+25, 516480)-Net in Base 5 — Upper bound on s

There is no (72, 97, 516481)-net in base 5, because

1 times m-reduction [i] would yield (72, 96, 516481)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 12 621788 383505 924388 201239 931240 575635 569078 805679 241139 972525 587025 > 5⁹⁶ [i]