MinT - Best Known (76−29, 76, s)-Nets in Base 8

Best Known (76−29, 76, s)-Nets in Base 8

(76−29, 76, 354)-Net over F₈ — Constructive and digital

Digital (47, 76, 354)-net over F₈, using

4 times m-reduction [i] based on digital (47, 80, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 40, 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]

(76−29, 76, 384)-Net in Base 8 — Constructive

(47, 76, 384)-net in base 8, using

trace code for nets [i] based on (9, 38, 192)-net in base 64, using
- 4 times m-reduction [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]

(76−29, 76, 489)-Net over F₈ — Digital

Digital (47, 76, 489)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷⁶, 489, F₈, 29) (dual of [489, 413, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁷⁶, 511, F₈, 29) (dual of [511, 435, 30]-code), using
  - the primitive expurgated narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [0,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(76−29, 76, 59467)-Net in Base 8 — Upper bound on s

There is no (47, 76, 59468)-net in base 8, because

1 times m-reduction [i] would yield (47, 75, 59468)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 53 926200 420553 677656 138909 088176 490849 521676 847045 905026 718680 531088 > 8⁷⁵ [i]