MinT - Best Known (123−34, 123, s)-Nets in Base 8

Best Known (123−34, 123, s)-Nets in Base 8

(123−34, 123, 562)-Net over F₈ — Constructive and digital

Digital (89, 123, 562)-net over F₈, using

1 times m-reduction [i] based on digital (89, 124, 562)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (23, 40, 208)-net over F₈, using
    - trace code for nets [i] based on digital (3, 20, 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 (49, 84, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 42, 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]

(123−34, 123, 612)-Net in Base 8 — Constructive

(89, 123, 612)-net in base 8, using

1 times m-reduction [i] based on (89, 124, 612)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (23, 40, 258)-net in base 8, using
    - trace code for nets [i] based on (3, 20, 129)-net in base 64, using
      - 1 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
        base change [i] based on digital (0, 18, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
  2. digital (49, 84, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 42, 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]

(123−34, 123, 4429)-Net over F₈ — Digital

Digital (89, 123, 4429)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹²³, 4429, F₈, 34) (dual of [4429, 4306, 35]-code), using
- 323 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 0, 0, 0, 1, 13 times 0, 1, 37 times 0, 1, 91 times 0, 1, 174 times 0) [i] based on linear OA(8¹¹⁷, 4100, F₈, 34) (dual of [4100, 3983, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(32) [i] based on
    1. linear OA(8¹¹⁷, 4096, F₈, 34) (dual of [4096, 3979, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(8¹¹³, 4096, F₈, 33) (dual of [4096, 3983, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [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]

(123−34, 123, 3507256)-Net in Base 8 — Upper bound on s

There is no (89, 123, 3507257)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1202 455905 323078 816301 924732 591100 351218 047481 037698 268543 634607 519572 057459 147516 741381 323719 712845 019573 706016 > 8¹²³ [i]