MinT - Best Known (139−55, 139, s)-Nets in Base 5

Best Known (139−55, 139, s)-Nets in Base 5

(139−55, 139, 252)-Net over F₅ — Constructive and digital

Digital (84, 139, 252)-net over F₅, using

9 times m-reduction [i] based on digital (84, 148, 252)-net over F₅, using
- trace code for nets [i] based on digital (10, 74, 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]

(139−55, 139, 312)-Net over F₅ — Digital

Digital (84, 139, 312)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−55, 139, 10186)-Net in Base 5 — Upper bound on s

There is no (84, 139, 10187)-net in base 5, because

1 times m-reduction [i] would yield (84, 138, 10187)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 874637 728620 716773 533439 353652 535510 334904 555068 352159 543723 490310 578406 671746 795767 621715 418885 > 5¹³⁸ [i]