MinT - Best Known (111, 111+56, s)-Nets in Base 8

Best Known (111, 111+56, s)-Nets in Base 8

(111, 111+56, 513)-Net over F₈ — Constructive and digital

Digital (111, 167, 513)-net over F₈, using

base reduction for projective spaces (embedding PG(83,64) in PG(166,8)) for nets [i] based on digital (28, 84, 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]

(111, 111+56, 576)-Net in Base 8 — Constructive

(111, 167, 576)-net in base 8, using

t-expansion [i] based on (108, 167, 576)-net in base 8, using
- 5 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]

(111, 111+56, 1711)-Net over F₈ — Digital

Digital (111, 167, 1711)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(111, 111+56, 392802)-Net in Base 8 — Upper bound on s

There is no (111, 167, 392803)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 546969 615203 156267 349017 068652 703765 299008 808875 533008 092788 025897 411148 610638 512506 616651 786208 368542 751985 495686 027005 587305 259574 158910 649423 688952 > 8¹⁶⁷ [i]