MinT - Best Known (76, 104, s)-Nets in Base 8

Best Known (76, 104, s)-Nets in Base 8

(76, 104, 562)-Net over F₈ — Constructive and digital

Digital (76, 104, 562)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (20, 34, 208)-net over F₈, using
  - trace code for nets [i] based on digital (3, 17, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
2. digital (42, 70, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 35, 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]

(76, 104, 644)-Net in Base 8 — Constructive

(76, 104, 644)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (14, 28, 130)-net over F₈, using
  - trace code for nets [i] based on digital (0, 14, 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. (48, 76, 514)-net in base 8, using
  - base change [i] based on digital (29, 57, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (29, 58, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 29, 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]

(76, 104, 4731)-Net over F₈ — Digital

Digital (76, 104, 4731)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰⁴, 4731, F₈, 28) (dual of [4731, 4627, 29]-code), using
- 624 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, 1, 187 times 0, 1, 290 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]

(76, 104, 4415982)-Net in Base 8 — Upper bound on s

There is no (76, 104, 4415983)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 8343 714035 425215 000169 909323 225933 100098 492325 842926 066451 189218 368389 026890 794616 654756 597549 > 8¹⁰⁴ [i]