MinT - Best Known (109−29, 109, s)-Nets in Base 8

Best Known (109−29, 109, s)-Nets in Base 8

(109−29, 109, 562)-Net over F₈ — Constructive and digital

Digital (80, 109, 562)-net over F₈, using

1 times m-reduction [i] based on digital (80, 110, 562)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 36, 208)-net over F₈, using
    - trace code for nets [i] based on digital (3, 18, 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 (44, 74, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 37, 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]

(109−29, 109, 674)-Net in Base 8 — Constructive

(80, 109, 674)-net in base 8, using

8¹ times duplication [i] based on (79, 108, 674)-net in base 8, 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. (49, 78, 514)-net in base 8, using
    - trace code for nets [i] based on (10, 39, 257)-net in base 64, using
      - 1 times m-reduction [i] based on (10, 40, 257)-net in base 64, using
        base change [i] based on digital (0, 30, 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]

(109−29, 109, 5306)-Net over F₈ — Digital

Digital (80, 109, 5306)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰⁹, 5306, F₈, 29) (dual of [5306, 5197, 30]-code), using
- 1198 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 5 times 0, 1, 23 times 0, 1, 66 times 0, 1, 148 times 0, 1, 250 times 0, 1, 327 times 0, 1, 372 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]

(109−29, 109, 7999326)-Net in Base 8 — Upper bound on s

There is no (80, 109, 7999327)-net in base 8, because

1 times m-reduction [i] would yield (80, 108, 7999327)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 34 175822 455586 662934 104039 209858 674255 717196 156536 387041 003340 405407 019729 133332 080697 788287 995403 > 8¹⁰⁸ [i]