MinT - Best Known (75, 104, s)-Nets in Base 8

Best Known (75, 104, s)-Nets in Base 8

(75, 104, 514)-Net over F₈ — Constructive and digital

Digital (75, 104, 514)-net over F₈, using

8² times duplication [i] based on digital (73, 102, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (16, 30, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 15, 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 (43, 72, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 36, 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]

(75, 104, 585)-Net in Base 8 — Constructive

(75, 104, 585)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 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. (61, 90, 576)-net in base 8, using
  - trace code for nets [i] based on (16, 45, 288)-net in base 64, using
    - 4 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]

(75, 104, 4133)-Net over F₈ — Digital

Digital (75, 104, 4133)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰⁴, 4133, F₈, 29) (dual of [4133, 4029, 30]-code), using
- 30 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 23 times 0) [i] based on linear OA(8¹⁰¹, 4100, F₈, 29) (dual of [4100, 3999, 30]-code), using
  - constructionÂ X applied to C_e(28) âŠ‚ C_e(27) [i] based on
    1. linear OA(8¹⁰¹, 4096, F₈, 29) (dual of [4096, 3995, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    2. linear OA(8⁹⁷, 4096, F₈, 28) (dual of [4096, 3999, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    3. linear OA(8⁰, 4, F₈, 0) (dual of [4, 4, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(8⁰, s, F₈, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(75, 104, 3806455)-Net in Base 8 — Upper bound on s

There is no (75, 104, 3806456)-net in base 8, because

1 times m-reduction [i] would yield (75, 103, 3806456)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1042 963671 900260 219213 429888 413954 668484 232913 626361 341963 567160 647203 840216 434898 215748 386426 > 8¹⁰³ [i]