MinT - Best Known (68, 95, s)-Nets in Base 8

Best Known (68, 95, s)-Nets in Base 8

(68, 95, 484)-Net over F₈ — Constructive and digital

Digital (68, 95, 484)-net over F₈, using

8¹ times duplication [i] based on digital (67, 94, 484)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 26, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 13, 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. digital (41, 68, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 34, 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]

(68, 95, 576)-Net in Base 8 — Constructive

(68, 95, 576)-net in base 8, using

7 times m-reduction [i] based on (68, 102, 576)-net in base 8, using
- trace code for nets [i] based on (17, 51, 288)-net in base 64, using
  - 5 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(68, 95, 3600)-Net over F₈ — Digital

Digital (68, 95, 3600)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹⁵, 3600, F₈, 27) (dual of [3600, 3505, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁹⁵, 4106, F₈, 27) (dual of [4106, 4011, 28]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8⁹⁴, 4105, F₈, 27) (dual of [4105, 4011, 28]-code), using
    - constructionÂ X applied to C_e(26) âŠ‚ C_e(24) [i] based on
      1. linear OA(8⁹³, 4096, F₈, 27) (dual of [4096, 4003, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
      2. linear OA(8⁸⁵, 4096, F₈, 25) (dual of [4096, 4011, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [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]

(68, 95, 2743735)-Net in Base 8 — Upper bound on s

There is no (68, 95, 2743736)-net in base 8, because

1 times m-reduction [i] would yield (68, 94, 2743736)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 7 770707 327629 033839 604271 250056 659288 336232 160608 740905 789772 232468 597945 201035 027434 > 8⁹⁴ [i]