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

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

(54, 54+20, 484)-Net over F₈ — Constructive and digital

Digital (54, 74, 484)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 20, 130)-net over F₈, using
  - trace code 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 (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, 54+20, 644)-Net in Base 8 — Constructive

(54, 74, 644)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (10, 20, 130)-net over F₈, using
  - trace code 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. (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, 54+20, 4218)-Net over F₈ — Digital

Digital (54, 74, 4218)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷⁴, 4218, F₈, 20) (dual of [4218, 4144, 21]-code), using
- 113 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 1, 8 times 0, 1, 27 times 0, 1, 72 times 0) [i] based on linear OA(8⁶⁹, 4100, F₈, 20) (dual of [4100, 4031, 21]-code), using
  - constructionÂ X applied to C_e(19) âŠ‚ C_e(18) [i] based on
    1. linear OA(8⁶⁹, 4096, F₈, 20) (dual of [4096, 4027, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [i]
    2. linear OA(8⁶⁵, 4096, F₈, 19) (dual of [4096, 4031, 20]-code), using an extension C_e(18) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,18], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 19 [i]
    3. linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(54, 54+20, 3117046)-Net in Base 8 — Upper bound on s

There is no (54, 74, 3117047)-net in base 8, because

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]