MinT - Best Known (65, 65+43, s)-Nets in Base 5

Best Known (65, 65+43, s)-Nets in Base 5

(65, 65+43, 252)-Net over F₅ — Constructive and digital

Digital (65, 108, 252)-net over F₅, using

2 times m-reduction [i] based on digital (65, 110, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 55, 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]

(65, 65+43, 253)-Net over F₅ — Digital

Digital (65, 108, 253)-net over F₅, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(5¹⁰⁸, 253, F₅, 2, 43) (dual of [(253, 2), 398, 44]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(5¹⁰⁶, 252, F₅, 2, 43) (dual of [(252, 2), 398, 44]-NRT-code), using
  - extracting embedded OOA [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]

(65, 65+43, 7888)-Net in Base 5 — Upper bound on s

There is no (65, 108, 7889)-net in base 5, because

1 times m-reduction [i] would yield (65, 107, 7889)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 616 563324 639046 927820 055298 740771 220438 271011 484328 967882 984809 422846 351925 > 5¹⁰⁷ [i]