MinT - Best Known (82, 82+28, s)-Nets in Base 8

Best Known (82, 82+28, s)-Nets in Base 8

(82, 82+28, 610)-Net over F₈ — Constructive and digital

Digital (82, 110, 610)-net over F₈, using

t-expansion [i] based on digital (81, 110, 610)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (24, 38, 256)-net over F₈, using
    - trace code for nets [i] based on digital (5, 19, 128)-net over F₆₄, using
      - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
        a function field by SÃ©mirat [i]
  2. digital (43, 72, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 36, 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]

(82, 82+28, 772)-Net in Base 8 — Constructive

(82, 110, 772)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (20, 34, 258)-net in base 8, using
  - trace code for nets [i] based on (3, 17, 129)-net in base 64, using
    - 4 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
      - base change [i] based on digital (0, 18, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
2. (48, 76, 514)-net in base 8, using
  - base change [i] based on digital (29, 57, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (29, 58, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 29, 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]

(82, 82+28, 8196)-Net over F₈ — Digital

Digital (82, 110, 8196)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹⁰, 8196, F₈, 28) (dual of [8196, 8086, 29]-code), using
- trace code [i] based on linear OA(64⁵⁵, 4098, F₆₄, 28) (dual of [4098, 4043, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(26) [i] based on
    1. linear OA(64⁵⁵, 4096, F₆₄, 28) (dual of [4096, 4041, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(64⁵³, 4096, F₆₄, 27) (dual of [4096, 4043, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 64²âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    3. linear OA(64⁰, 2, F₆₄, 0) (dual of [2, 2, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(64⁰, s, F₆₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(82, 82+28, large)-Net in Base 8 — Upper bound on s

There is no (82, 110, large)-net in base 8, because

26 times m-reduction [i] would yield (82, 84, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]