MinT - Best Known (54, 88, s)-Nets in Base 8

Best Known (54, 88, s)-Nets in Base 8

(54, 88, 354)-Net over F₈ — Constructive and digital

Digital (54, 88, 354)-net over F₈, using

6 times m-reduction [i] based on digital (54, 94, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 47, 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]

(54, 88, 384)-Net in Base 8 — Constructive

(54, 88, 384)-net in base 8, using

trace code for nets [i] based on (10, 44, 192)-net in base 64, using
- 5 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
  - base change [i] based on digital (3, 42, 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]

(54, 88, 504)-Net over F₈ — Digital

Digital (54, 88, 504)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁸, 504, F₈, 34) (dual of [504, 416, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁸⁸, 511, F₈, 34) (dual of [511, 423, 35]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [i]

(54, 88, 48481)-Net in Base 8 — Upper bound on s

There is no (54, 88, 48482)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 29 652065 352558 102267 106959 048183 586124 430038 222009 505669 308939 192056 378953 773646 > 8⁸⁸ [i]