MinT - Best Known (92, 92+37, s)-Nets in Base 8

Best Known (92, 92+37, s)-Nets in Base 8

(92, 92+37, 514)-Net over F₈ — Constructive and digital

Digital (92, 129, 514)-net over F₈, using

1 times m-reduction [i] based on digital (92, 130, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 40, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 20, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
  2. digital (52, 90, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 45, 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]

(92, 92+37, 600)-Net in Base 8 — Constructive

(92, 129, 600)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (3, 21, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. (71, 108, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 54, 288)-net in base 64, using
    - 2 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]

(92, 92+37, 3968)-Net over F₈ — Digital

Digital (92, 129, 3968)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²⁹, 3968, F₈, 37) (dual of [3968, 3839, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹²⁹, 4096, F₈, 37) (dual of [4096, 3967, 38]-code), using
  - an extension C_e(36) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,36], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 37 [i]

(92, 92+37, 2851041)-Net in Base 8 — Upper bound on s

There is no (92, 129, 2851042)-net in base 8, because

1 times m-reduction [i] would yield (92, 128, 2851042)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 39 402078 272492 058288 080742 364385 046169 651970 141835 730512 253895 463569 450983 352717 950793 121630 456162 425201 590574 852080 > 8¹²⁸ [i]