MinT - Best Known (85, 119, s)-Nets in Base 8

Best Known (85, 119, s)-Nets in Base 8

(85, 119, 514)-Net over F₈ — Constructive and digital

Digital (85, 119, 514)-net over F₈, using

1 times m-reduction [i] based on digital (85, 120, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (19, 36, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 18, 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 (49, 84, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 42, 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]

(85, 119, 585)-Net in Base 8 — Constructive

(85, 119, 585)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (0, 17, 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. (68, 102, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 51, 288)-net in base 64, using
    - 5 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]

(85, 119, 3889)-Net over F₈ — Digital

Digital (85, 119, 3889)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹¹⁹, 3889, F₈, 34) (dual of [3889, 3770, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹¹⁹, 4105, F₈, 34) (dual of [4105, 3986, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(30) [i] based on
    1. linear OA(8¹¹⁷, 4096, F₈, 34) (dual of [4096, 3979, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. 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]
    3. linear OA(8², 9, F₈, 2) (dual of [9, 7, 3]-code or 9-arc in PG(1,8)), using
      - extended Reedâ€“Solomon code RS_e(7,8) [i]
      - Hamming code H(2,8) [i]

(85, 119, 2150180)-Net in Base 8 — Upper bound on s

There is no (85, 119, 2150181)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 293568 403942 857093 145231 793408 385446 552285 202492 583908 935183 252790 637636 644533 074423 994111 862165 092746 218584 > 8¹¹⁹ [i]