MinT - Best Known (78, 78+55, s)-Nets in Base 5

Best Known (78, 78+55, s)-Nets in Base 5

(78, 78+55, 252)-Net over F₅ — Constructive and digital

Digital (78, 133, 252)-net over F₅, using

3 times m-reduction [i] based on digital (78, 136, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 68, 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]

(78, 78+55, 257)-Net over F₅ — Digital

Digital (78, 133, 257)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+55, 7117)-Net in Base 5 — Upper bound on s

There is no (78, 133, 7118)-net in base 5, because

1 times m-reduction [i] would yield (78, 132, 7118)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 183 975831 724883 649831 737560 910581 387702 021193 707402 435092 249870 625272 389533 209026 121732 442201 > 5¹³² [i]