MinT - Best Known (109−31, 109, s)-Nets in Base 8

Best Known (109−31, 109, s)-Nets in Base 8

(109−31, 109, 514)-Net over F₈ — Constructive and digital

Digital (78, 109, 514)-net over F₈, using

8¹ times duplication [i] based on digital (77, 108, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (17, 32, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 16, 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 (45, 76, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 38, 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]

(109−31, 109, 585)-Net in Base 8 — Constructive

(78, 109, 585)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. (63, 94, 576)-net in base 8, using
  - trace code for nets [i] based on (16, 47, 288)-net in base 64, using
    - 2 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]

(109−31, 109, 3831)-Net over F₈ — Digital

Digital (78, 109, 3831)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰⁹, 3831, F₈, 31) (dual of [3831, 3722, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹⁰⁹, 4096, F₈, 31) (dual of [4096, 3987, 32]-code), using
  - an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]

(109−31, 109, 2916857)-Net in Base 8 — Upper bound on s

There is no (78, 109, 2916858)-net in base 8, because

1 times m-reduction [i] would yield (78, 108, 2916858)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 34 175959 369465 418841 709257 274313 029412 374530 052535 209668 119321 040300 549917 803218 405263 282452 888696 > 8¹⁰⁸ [i]