MinT - Best Known (172−49, 172, s)-Nets in Base 8

Best Known (172−49, 172, s)-Nets in Base 8

(172−49, 172, 1026)-Net over F₈ — Constructive and digital

Digital (123, 172, 1026)-net over F₈, using

t-expansion [i] based on digital (114, 172, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 86, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(172−49, 172, 4635)-Net over F₈ — Digital

Digital (123, 172, 4635)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−49, 172, 3808720)-Net in Base 8 — Upper bound on s

There is no (123, 172, 3808721)-net in base 8, because

1 times m-reduction [i] would yield (123, 171, 3808721)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 26815 629440 494424 170389 401498 903605 201032 492478 785432 508525 885452 309030 728261 571220 306445 284580 839615 747791 193194 539889 649993 492682 869573 151197 171720 498456 > 8¹⁷¹ [i]