MinT - Best Known (145−49, 145, s)-Nets in Base 8

Best Known (145−49, 145, s)-Nets in Base 8

(145−49, 145, 399)-Net over F₈ — Constructive and digital

Digital (96, 145, 399)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 33, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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 (63, 112, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 56, 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]

(145−49, 145, 576)-Net in Base 8 — Constructive

(96, 145, 576)-net in base 8, using

7 times m-reduction [i] based on (96, 152, 576)-net in base 8, using
- trace code for nets [i] based on (20, 76, 288)-net in base 64, using
  - 1 times m-reduction [i] based on (20, 77, 288)-net in base 64, using
    - base change [i] based on digital (9, 66, 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]

(145−49, 145, 1455)-Net over F₈ — Digital

Digital (96, 145, 1455)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(145−49, 145, 367102)-Net in Base 8 — Upper bound on s

There is no (96, 145, 367103)-net in base 8, because

1 times m-reduction [i] would yield (96, 144, 367103)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11091 276860 840745 870524 409196 623875 983665 636359 715209 696575 425731 848443 247559 285018 047042 458673 477898 482270 157385 316635 664798 483769 > 8¹⁴⁴ [i]