MinT - Best Known (81, 81+30, s)-Nets in Base 8

Best Known (81, 81+30, s)-Nets in Base 8

(81, 81+30, 562)-Net over F₈ — Constructive and digital

Digital (81, 111, 562)-net over F₈, using

1 times m-reduction [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]

(81, 81+30, 644)-Net in Base 8 — Constructive

(81, 111, 644)-net in base 8, using

8¹ times duplication [i] based on (80, 110, 644)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 30, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 15, 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. (50, 80, 514)-net in base 8, using
    - base change [i] based on digital (30, 60, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 30, 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]

(81, 81+30, 4794)-Net over F₈ — Digital

Digital (81, 111, 4794)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹¹, 4794, F₈, 30) (dual of [4794, 4683, 31]-code), using
- 688 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 6 times 0, 1, 38 times 0, 1, 117 times 0, 1, 221 times 0, 1, 301 times 0) [i] based on linear OA(8¹⁰⁵, 4100, F₈, 30) (dual of [4100, 3995, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(28) [i] based on
    1. linear OA(8¹⁰⁵, 4096, F₈, 30) (dual of [4096, 3991, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [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(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]

(81, 81+30, 4421133)-Net in Base 8 — Upper bound on s

There is no (81, 111, 4421134)-net in base 8, because

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]