MinT - Best Known (52, 52+33, s)-Nets in Base 8

Best Known (52, 52+33, s)-Nets in Base 8

(52, 52+33, 354)-Net over F₈ — Constructive and digital

Digital (52, 85, 354)-net over F₈, using

5 times m-reduction [i] based on digital (52, 90, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 45, 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]

(52, 52+33, 384)-Net in Base 8 — Constructive

(52, 85, 384)-net in base 8, using

8¹ times duplication [i] based on (51, 84, 384)-net in base 8, using
- trace code for nets [i] based on (9, 42, 192)-net in base 64, using
  - base change [i] based on digital (3, 36, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(52, 52+33, 480)-Net over F₈ — Digital

Digital (52, 85, 480)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁵, 480, F₈, 33) (dual of [480, 395, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁸⁵, 511, F₈, 33) (dual of [511, 426, 34]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]

(52, 52+33, 53528)-Net in Base 8 — Upper bound on s

There is no (52, 85, 53529)-net in base 8, because

1 times m-reduction [i] would yield (52, 84, 53529)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 7238 444426 643139 988693 807785 933431 531534 230461 456633 437213 016096 220816 410623 > 8⁸⁴ [i]