MinT - Best Known (89−24, 89, s)-Nets in Base 8

Best Known (89−24, 89, s)-Nets in Base 8

(89−24, 89, 514)-Net over F₈ — Constructive and digital

Digital (65, 89, 514)-net over F₈, using

1 times m-reduction [i] based on digital (65, 90, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 26, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 13, 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 (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]

(89−24, 89, 644)-Net in Base 8 — Constructive

(65, 89, 644)-net in base 8, using

8¹ times duplication [i] based on (64, 88, 644)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. digital (12, 24, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 12, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. (40, 64, 514)-net in base 8, using
    - base change [i] based on digital (24, 48, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 24, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(89−24, 89, 4367)-Net over F₈ — Digital

Digital (65, 89, 4367)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁸⁹, 4367, F₈, 24) (dual of [4367, 4278, 25]-code), using
- 267 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 24 times 0, 1, 70 times 0, 1, 164 times 0) [i] based on linear OA(8⁸⁴, 4095, F₈, 24) (dual of [4095, 4011, 25]-code), using
  - 1 times truncation [i] 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]

(89−24, 89, 3768599)-Net in Base 8 — Upper bound on s

There is no (65, 89, 3768600)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 237 142840 769452 193242 816603 759632 299426 776788 699246 538932 104565 668523 518630 891246 > 8⁸⁹ [i]