MinT - Best Known (104−41, 104, s)-Nets in Base 5

Best Known (104−41, 104, s)-Nets in Base 5

(104−41, 104, 252)-Net over F₅ — Constructive and digital

Digital (63, 104, 252)-net over F₅, using

2 times m-reduction [i] based on digital (63, 106, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 53, 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]

(104−41, 104, 253)-Net over F₅ — Digital

Digital (63, 104, 253)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(5¹⁰⁴, 253, F₅, 2, 41) (dual of [(253, 2), 402, 42]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(5¹⁰², 252, F₅, 2, 41) (dual of [(252, 2), 402, 42]-NRT-code), using
  - extracting embedded OOA [i] based on digital (61, 102, 252)-net over F₅, using
    - trace code for nets [i] based on digital (10, 51, 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]

(104−41, 104, 8244)-Net in Base 5 — Upper bound on s

There is no (63, 104, 8245)-net in base 5, because

1 times m-reduction [i] would yield (63, 103, 8245)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 986599 046933 903909 243009 989614 138886 780830 086049 422317 691947 005715 747505 > 5¹⁰³ [i]