MinT - Best Known (78, 78+44, s)-Nets in Base 8

Best Known (78, 78+44, s)-Nets in Base 8

(78, 78+44, 354)-Net over F₈ — Constructive and digital

Digital (78, 122, 354)-net over F₈, using

20 times m-reduction [i] based on digital (78, 142, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 71, 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]

(78, 78+44, 516)-Net in Base 8 — Constructive

(78, 122, 516)-net in base 8, using

8² times duplication [i] based on (76, 120, 516)-net in base 8, using
- base change [i] based on digital (46, 90, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 45, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(78, 78+44, 902)-Net over F₈ — Digital

Digital (78, 122, 902)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+44, 131751)-Net in Base 8 — Upper bound on s

There is no (78, 122, 131752)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 150 327463 791808 768749 302796 473428 076635 664493 961510 616298 081864 821284 893379 636942 579666 929653 416872 876748 826746 > 8¹²² [i]