MinT - Best Known (124−45, 124, s)-Nets in Base 8

Best Known (124−45, 124, s)-Nets in Base 8

(124−45, 124, 354)-Net over F₈ — Constructive and digital

Digital (79, 124, 354)-net over F₈, using

20 times m-reduction [i] based on digital (79, 144, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 72, 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]

(124−45, 124, 516)-Net in Base 8 — Constructive

(79, 124, 516)-net in base 8, using

base change [i] based on digital (48, 93, 516)-net over F₁₆, using
- 1 times m-reduction [i] based on digital (48, 94, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 47, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(124−45, 124, 887)-Net over F₈ — Digital

Digital (79, 124, 887)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(124−45, 124, 144813)-Net in Base 8 — Upper bound on s

There is no (79, 124, 144814)-net in base 8, because

1 times m-reduction [i] would yield (79, 123, 144814)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1202 603417 882057 722669 710152 291584 101381 761673 981095 994528 133738 625923 976681 508771 023095 874763 557248 072506 894168 > 8¹²³ [i]