MinT - Best Known (107, 107+52, s)-Nets in Base 8

Best Known (107, 107+52, s)-Nets in Base 8

(107, 107+52, 513)-Net over F₈ — Constructive and digital

Digital (107, 159, 513)-net over F₈, using

base reduction for projective spaces (embedding PG(79,64) in PG(158,8)) for nets [i] based on digital (28, 80, 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]

(107, 107+52, 576)-Net in Base 8 — Constructive

(107, 159, 576)-net in base 8, using

t-expansion [i] based on (105, 159, 576)-net in base 8, using
- 9 times m-reduction [i] based on (105, 168, 576)-net in base 8, using
  - trace code for nets [i] based on (21, 84, 288)-net in base 64, using
    - base change [i] based on digital (9, 72, 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]

(107, 107+52, 1880)-Net over F₈ — Digital

Digital (107, 159, 1880)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(107, 107+52, 502254)-Net in Base 8 — Upper bound on s

There is no (107, 159, 502255)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 390234 375436 509398 531728 121462 901150 131118 791431 975776 751351 364650 729084 182613 390646 086198 044547 054899 745510 579189 240798 282030 499668 436951 343478 > 8¹⁵⁹ [i]