MinT - Best Known (74, 74+28, s)-Nets in Base 8

Best Known (74, 74+28, s)-Nets in Base 8

(74, 74+28, 514)-Net over F₈ — Constructive and digital

Digital (74, 102, 514)-net over F₈, using

t-expansion [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]

(74, 74+28, 585)-Net in Base 8 — Constructive

(74, 102, 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. (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]

(74, 74+28, 4250)-Net over F₈ — Digital

Digital (74, 102, 4250)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰², 4250, F₈, 28) (dual of [4250, 4148, 29]-code), using
- 145 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 12 times 0, 1, 36 times 0, 1, 90 times 0) [i] based on linear OA(8⁹⁷, 4100, F₈, 28) (dual of [4100, 4003, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(26) [i] based on
    1. 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]
    2. linear OA(8⁹³, 4096, F₈, 27) (dual of [4096, 4003, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [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]

(74, 74+28, 3281060)-Net in Base 8 — Upper bound on s

There is no (74, 102, 3281061)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 130 370768 254365 395306 814061 816360 707482 140446 694604 606874 555980 476467 224209 764189 925116 090032 > 8¹⁰² [i]