MinT - Best Known (89, 89+36, s)-Nets in Base 8

Best Known (89, 89+36, s)-Nets in Base 8

(89, 89+36, 514)-Net over F₈ — Constructive and digital

Digital (89, 125, 514)-net over F₈, using

1 times m-reduction [i] based on digital (89, 126, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (20, 38, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 19, 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 (51, 88, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 44, 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]

(89, 89+36, 590)-Net in Base 8 — Constructive

(89, 125, 590)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. (70, 106, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 53, 288)-net in base 64, using
    - 3 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]

(89, 89+36, 3781)-Net over F₈ — Digital

Digital (89, 125, 3781)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²⁵, 3781, F₈, 36) (dual of [3781, 3656, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹²⁵, 4096, F₈, 36) (dual of [4096, 3971, 37]-code), using
  - an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]

(89, 89+36, 2015987)-Net in Base 8 — Upper bound on s

There is no (89, 125, 2015988)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 76957 260457 366771 052252 764871 755870 723399 596767 611425 727473 327963 366615 141807 829753 227288 503911 142829 680969 409721 > 8¹²⁵ [i]