MinT - Best Known (91, 91+45, s)-Nets in Base 8

Best Known (91, 91+45, s)-Nets in Base 8

(91, 91+45, 400)-Net over F₈ — Constructive and digital

Digital (91, 136, 400)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 32, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (59, 104, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 52, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(91, 91+45, 576)-Net in Base 8 — Constructive

(91, 136, 576)-net in base 8, using

t-expansion [i] based on (89, 136, 576)-net in base 8, using
- 4 times m-reduction [i] based on (89, 140, 576)-net in base 8, using
  - trace code for nets [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(91, 91+45, 1547)-Net over F₈ — Digital

Digital (91, 136, 1547)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(91, 91+45, 450225)-Net in Base 8 — Upper bound on s

There is no (91, 136, 450226)-net in base 8, because

1 times m-reduction [i] would yield (91, 135, 450226)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 82 632781 708798 153231 559579 871439 800297 976584 588895 490214 283665 826527 031341 716009 607485 632724 021271 617222 548355 648562 976144 > 8¹³⁵ [i]