MinT - Best Known (141−43, 141, s)-Nets in Base 8

Best Known (141−43, 141, s)-Nets in Base 8

(141−43, 141, 513)-Net over F₈ — Constructive and digital

Digital (98, 141, 513)-net over F₈, using

base reduction for projective spaces (embedding PG(70,64) in PG(140,8)) for nets [i] based on digital (28, 71, 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]

(141−43, 141, 576)-Net in Base 8 — Constructive

(98, 141, 576)-net in base 8, using

t-expansion [i] based on (97, 141, 576)-net in base 8, using
- 13 times m-reduction [i] based on (97, 154, 576)-net in base 8, using
  - trace code for nets [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]

(141−43, 141, 2559)-Net over F₈ — Digital

Digital (98, 141, 2559)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(141−43, 141, 1300139)-Net in Base 8 — Upper bound on s

There is no (98, 141, 1300140)-net in base 8, because

1 times m-reduction [i] would yield (98, 140, 1300140)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 707688 118186 663043 927107 757392 623276 207628 915516 941455 429909 436313 047885 320447 541434 112010 922817 273692 102391 742347 439237 751084 > 8¹⁴⁰ [i]