MinT - Best Known (67, 67+27, s)-Nets in Base 8

Best Known (67, 67+27, s)-Nets in Base 8

(67, 67+27, 484)-Net over F₈ — Constructive and digital

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]

(67, 67+27, 576)-Net in Base 8 — Constructive

(67, 94, 576)-net in base 8, using

t-expansion [i] based on (65, 94, 576)-net in base 8, using
- 4 times m-reduction [i] based on (65, 98, 576)-net in base 8, using
  - trace code for nets [i] based on (16, 49, 288)-net in base 64, using
    - base change [i] based on digital (9, 42, 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]

(67, 67+27, 3312)-Net over F₈ — Digital

Digital (67, 94, 3312)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹⁴, 3312, F₈, 27) (dual of [3312, 3218, 28]-code), using
- discarding factors / shortening the dual code 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]

(67, 67+27, 2338156)-Net in Base 8 — Upper bound on s

There is no (67, 94, 2338157)-net in base 8, because

1 times m-reduction [i] would yield (67, 93, 2338157)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 971339 114041 507515 808700 951682 339734 036586 177521 609542 256809 532259 784180 019055 400296 > 8⁹³ [i]