MinT - Best Known (57, 57+21, s)-Nets in Base 8

Best Known (57, 57+21, s)-Nets in Base 8

(57, 57+21, 514)-Net over F₈ — Constructive and digital

Digital (57, 78, 514)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (12, 22, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 11, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [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]

(57, 57+21, 674)-Net in Base 8 — Constructive

(57, 78, 674)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (12, 22, 160)-net over F₈, using
  - trace code for nets [i] based on digital (1, 11, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [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]

(57, 57+21, 4279)-Net over F₈ — Digital

Digital (57, 78, 4279)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷⁸, 4279, F₈, 21) (dual of [4279, 4201, 22]-code), using
- 174 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 14 times 0, 1, 43 times 0, 1, 110 times 0) [i] based on linear OA(8⁷³, 4100, F₈, 21) (dual of [4100, 4027, 22]-code), using
  - constructionÂ X applied to C_e(20) âŠ‚ C_e(19) [i] based on
    1. linear OA(8⁷³, 4096, F₈, 21) (dual of [4096, 4023, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
    2. 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]
    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]

(57, 57+21, 5816620)-Net in Base 8 — Upper bound on s

There is no (57, 78, 5816621)-net in base 8, because

1 times m-reduction [i] would yield (57, 77, 5816621)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3450 876864 786824 577542 150113 732350 422868 159242 363983 559112 197518 106346 > 8⁷⁷ [i]