MinT - Best Known (82, 82+31, s)-Nets in Base 8

Best Known (82, 82+31, s)-Nets in Base 8

(82, 82+31, 562)-Net over F₈ — Constructive and digital

Digital (82, 113, 562)-net over F₈, using

8¹ times duplication [i] based on 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]

(82, 82+31, 612)-Net in Base 8 — Constructive

(82, 113, 612)-net in base 8, using

8¹ times duplication [i] based on (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]

(82, 82+31, 4376)-Net over F₈ — Digital

Digital (82, 113, 4376)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹³, 4376, F₈, 31) (dual of [4376, 4263, 32]-code), using
- 268 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (1, 9 times 0, 1, 62 times 0, 1, 194 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]

(82, 82+31, 5078549)-Net in Base 8 — Upper bound on s

There is no (82, 113, 5078550)-net in base 8, because

1 times m-reduction [i] would yield (82, 112, 5078550)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 139984 098773 923369 881686 049526 325911 055674 618294 261263 868867 197726 003955 611176 742689 530739 344868 462856 > 8¹¹² [i]