MinT - Best Known (173−85, 173, s)-Nets in Base 8

Best Known (173−85, 173, s)-Nets in Base 8

(173−85, 173, 194)-Net over F₈ — Constructive and digital

Digital (88, 173, 194)-net over F₈, using

t-expansion [i] based on digital (85, 173, 194)-net over F₈, using
- net from sequence [i] based on digital (85, 193)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 85 and N(F) ≥ 194, using
    - F₆ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(173−85, 173, 225)-Net in Base 8 — Constructive

(88, 173, 225)-net in base 8, using

8¹ times duplication [i] based on (87, 172, 225)-net in base 8, using
- t-expansion [i] based on (83, 172, 225)-net in base 8, using
  - base change [i] based on digital (40, 129, 225)-net over F₁₆, using
    - net from sequence [i] based on digital (40, 224)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 40 and N(F) ≥ 225, using
        a function field by GarcÃa and Quoos [i]

(173−85, 173, 296)-Net over F₈ — Digital

Digital (88, 173, 296)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(173−85, 173, 11751)-Net in Base 8 — Upper bound on s

There is no (88, 173, 11752)-net in base 8, because

1 times m-reduction [i] would yield (88, 172, 11752)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 214877 417298 322524 197696 300346 143513 108704 898783 213640 449534 793325 919986 311937 763810 552415 574217 323085 201065 187619 954640 237276 163752 289364 436021 331881 846696 > 8¹⁷² [i]