MinT - Best Known (60, 85, s)-Nets in Base 8

Best Known (60, 85, s)-Nets in Base 8

(60, 85, 399)-Net over F₈ — Constructive and digital

Digital (60, 85, 399)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 21, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]
2. digital (39, 64, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 32, 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]

(60, 85, 576)-Net in Base 8 — Constructive

(60, 85, 576)-net in base 8, using

3 times m-reduction [i] based on (60, 88, 576)-net in base 8, using
- trace code for nets [i] based on (16, 44, 288)-net in base 64, using
  - 5 times m-reduction [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]

(60, 85, 2663)-Net over F₈ — Digital

Digital (60, 85, 2663)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁵, 2663, F₈, 25) (dual of [2663, 2578, 26]-code), using
- discarding factors / shortening the dual code based on 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]

(60, 85, 1584496)-Net in Base 8 — Upper bound on s

There is no (60, 85, 1584497)-net in base 8, because

1 times m-reduction [i] would yield (60, 84, 1584497)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 7237 028775 322376 165100 365498 501625 736421 733329 678642 869492 625440 265343 226516 > 8⁸⁴ [i]