MinT - Best Known (65, 65+42, s)-Nets in Base 25

Best Known (65, 65+42, s)-Nets in Base 25

(65, 65+42, 378)-Net over F₂₅ — Constructive and digital

Digital (65, 107, 378)-net over F₂₅, using

1 times m-reduction [i] based on digital (65, 108, 378)-net over F₂₅, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (10, 24, 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]
  2. digital (10, 31, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)
  3. digital (10, 53, 126)-net over F₂₅, using
    - net from sequence [i] based on digital (10, 125)-sequence over F₂₅ (see above)

(65, 65+42, 3013)-Net over F₂₅ — Digital

Digital (65, 107, 3013)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+42, 4798632)-Net in Base 25 — Upper bound on s

There is no (65, 107, 4798633)-net in base 25, because

the generalized Rao bound for nets shows that 25^mÂ â‰¥ 379822 856516 023635 520274 096361 786989 571940 956309 347178 481258 536925 061265 367557 206399 610169 678266 492604 114630 407651 559765 899675 334249 250106 412443 828025 > 25¹⁰⁷ [i]