MinT - Best Known (73−25, 73, s)-Nets in Base 8

Best Known (73−25, 73, s)-Nets in Base 8

(73−25, 73, 354)-Net over F₈ — Constructive and digital

Digital (48, 73, 354)-net over F₈, using

9 times m-reduction [i] based on digital (48, 82, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 41, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(73−25, 73, 518)-Net in Base 8 — Constructive

(48, 73, 518)-net in base 8, using

8¹ times duplication [i] based on (47, 72, 518)-net in base 8, using
- base change [i] based on digital (29, 54, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 27, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(73−25, 73, 794)-Net over F₈ — Digital

Digital (48, 73, 794)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(73−25, 73, 198055)-Net in Base 8 — Upper bound on s

There is no (48, 73, 198056)-net in base 8, because

1 times m-reduction [i] would yield (48, 72, 198056)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 105313 825072 924091 231045 137591 699550 308738 938126 015359 162494 337484 > 8⁷² [i]