MinT - Best Known (100, 100+45, s)-Nets in Base 8

Best Known (100, 100+45, s)-Nets in Base 8

(100, 100+45, 513)-Net over F₈ — Constructive and digital

Digital (100, 145, 513)-net over F₈, using

base reduction for projective spaces (embedding PG(72,64) in PG(144,8)) for nets [i] based on digital (28, 73, 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]

(100, 100+45, 576)-Net in Base 8 — Constructive

(100, 145, 576)-net in base 8, using

13 times m-reduction [i] based on (100, 158, 576)-net in base 8, using
- trace code for nets [i] based on (21, 79, 288)-net in base 64, using
  - 5 times m-reduction [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]

(100, 100+45, 2355)-Net over F₈ — Digital

Digital (100, 145, 2355)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(100, 100+45, 1054104)-Net in Base 8 — Upper bound on s

There is no (100, 145, 1054105)-net in base 8, because

1 times m-reduction [i] would yield (100, 144, 1054105)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11090 705163 249587 223128 739120 682318 040429 988174 049882 125997 204376 714542 376038 095435 218179 313394 404134 650576 998226 234093 338659 990196 > 8¹⁴⁴ [i]