MinT - Best Known (54, 54+21, s)-Nets in Base 8

Best Known (54, 54+21, s)-Nets in Base 8

(54, 54+21, 419)-Net over F₈ — Constructive and digital

Digital (54, 75, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 19, 65)-net over F₈, using
  - base reduction for projective spaces (embedding PG(9,64) in PG(18,8)) for nets [i] based on digital (0, 10, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. digital (35, 56, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 28, 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, 54+21, 579)-Net in Base 8 — Constructive

(54, 75, 579)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (9, 19, 65)-net over F₈, using
  - base reduction for projective spaces (embedding PG(9,64) in PG(18,8)) for nets [i] based on digital (0, 10, 65)-net over F₆₄, using
    - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
      - Niederreiter sequence [i]
2. (35, 56, 514)-net in base 8, using
  - base change [i] based on digital (21, 42, 514)-net over F₁₆, using
    - trace code for nets [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, 54+21, 3716)-Net over F₈ — Digital

Digital (54, 75, 3716)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷⁵, 3716, F₈, 21) (dual of [3716, 3641, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁷⁵, 4107, F₈, 21) (dual of [4107, 4032, 22]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8⁷⁴, 4106, F₈, 21) (dual of [4106, 4032, 22]-code), using
    - constructionÂ X applied to C([0,10]) âŠ‚ C([0,9]) [i] based on
      1. linear OA(8⁷³, 4097, F₈, 21) (dual of [4097, 4024, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
      2. linear OA(8⁶⁵, 4097, F₈, 19) (dual of [4097, 4032, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 8⁸âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      3. linear OA(8¹, 9, F₈, 1) (dual of [9, 8, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(54, 54+21, 3117046)-Net in Base 8 — Upper bound on s

There is no (54, 75, 3117047)-net in base 8, because

1 times m-reduction [i] would yield (54, 74, 3117047)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 739988 185598 489419 518320 522008 221829 801436 191718 999029 149696 438906 > 8⁷⁴ [i]