MinT - Best Known (81, 81+31, s)-Nets in Base 8

Best Known (81, 81+31, s)-Nets in Base 8

(81, 81+31, 562)-Net over F₈ — Constructive and digital

Digital (81, 112, 562)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (21, 36, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 18, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
2. digital (45, 76, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 38, 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]

(81, 81+31, 612)-Net in Base 8 — Constructive

(81, 112, 612)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. (21, 36, 258)-net in base 8, using
  - trace code for nets [i] based on (3, 18, 129)-net in base 64, using
    - 3 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. digital (45, 76, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 38, 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]

(81, 81+31, 4180)-Net over F₈ — Digital

Digital (81, 112, 4180)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹², 4180, F₈, 31) (dual of [4180, 4068, 32]-code), using
- 73 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 9 times 0, 1, 62 times 0) [i] based on linear OA(8¹¹⁰, 4105, F₈, 31) (dual of [4105, 3995, 32]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(28) [i] based on
    1. linear OA(8¹⁰⁹, 4096, F₈, 31) (dual of [4096, 3987, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    2. linear OA(8¹⁰¹, 4096, F₈, 29) (dual of [4096, 3995, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [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]

(81, 81+31, 4421133)-Net in Base 8 — Upper bound on s

There is no (81, 112, 4421134)-net in base 8, because

1 times m-reduction [i] would yield (81, 111, 4421134)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 17498 049172 675162 513761 723766 879096 227985 738575 819078 927611 946304 842685 598407 412674 861795 468406 032576 > 8¹¹¹ [i]