MinT - Best Known (69−26, 69, s)-Nets in Base 8

Best Known (69−26, 69, s)-Nets in Base 8

(69−26, 69, 354)-Net over F₈ — Constructive and digital

Digital (43, 69, 354)-net over F₈, using

3 times m-reduction [i] based on digital (43, 72, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 36, 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]

(69−26, 69, 384)-Net in Base 8 — Constructive

(43, 69, 384)-net in base 8, using

1 times m-reduction [i] based on (43, 70, 384)-net in base 8, using
- trace code for nets [i] based on (8, 35, 192)-net in base 64, using
  - base change [i] based on digital (3, 30, 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]

(69−26, 69, 494)-Net over F₈ — Digital

Digital (43, 69, 494)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁶⁹, 494, F₈, 26) (dual of [494, 425, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁶⁹, 511, F₈, 26) (dual of [511, 442, 27]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]

(69−26, 69, 50299)-Net in Base 8 — Upper bound on s

There is no (43, 69, 50300)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 205 712085 352008 522522 613714 603536 634662 967886 735241 651773 598636 > 8⁶⁹ [i]