MinT - Best Known (163−54, 163, s)-Nets in Base 8

Best Known (163−54, 163, s)-Nets in Base 8

(163−54, 163, 513)-Net over F₈ — Constructive and digital

Digital (109, 163, 513)-net over F₈, using

base reduction for projective spaces (embedding PG(81,64) in PG(162,8)) for nets [i] based on digital (28, 82, 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]

(163−54, 163, 576)-Net in Base 8 — Constructive

(109, 163, 576)-net in base 8, using

t-expansion [i] based on (108, 163, 576)-net in base 8, using
- 9 times m-reduction [i] based on (108, 172, 576)-net in base 8, using
  - trace code for nets [i] based on (22, 86, 288)-net in base 64, using
    - 5 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
      - base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
        net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(163−54, 163, 1789)-Net over F₈ — Digital

Digital (109, 163, 1789)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(163−54, 163, 441855)-Net in Base 8 — Upper bound on s

There is no (109, 163, 441856)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1598 424252 006576 614540 174205 168958 010859 190113 923617 237948 557167 664975 125302 395633 756041 709353 061354 032805 057592 576552 471126 209351 459714 662506 964129 > 8¹⁶³ [i]