MinT - Best Known (54−20, 54, s)-Nets in Base 8

Best Known (54−20, 54, s)-Nets in Base 8

(54−20, 54, 354)-Net over F₈ — Constructive and digital

Digital (34, 54, 354)-net over F₈, using

trace code for nets [i] based on digital (7, 27, 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]

(54−20, 54, 482)-Net over F₈ — Digital

Digital (34, 54, 482)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵⁴, 482, F₈, 20) (dual of [482, 428, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁵⁴, 511, F₈, 20) (dual of [511, 457, 21]-code), using
  - the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]

(54−20, 54, 514)-Net in Base 8 — Constructive

(34, 54, 514)-net in base 8, using

trace code for nets [i] based on (7, 27, 257)-net in base 64, using
- 1 times m-reduction [i] based on (7, 28, 257)-net in base 64, using
  - base change [i] based on digital (0, 21, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(54−20, 54, 48698)-Net in Base 8 — Upper bound on s

There is no (34, 54, 48699)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5 847179 872269 287379 533274 927729 289703 322157 654371 > 8⁵⁴ [i]