MinT - Best Known (49, 79, s)-Nets in Base 8

Best Known (49, 79, s)-Nets in Base 8

(49, 79, 354)-Net over F₈ — Constructive and digital

Digital (49, 79, 354)-net over F₈, using

5 times m-reduction [i] based on 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]

(49, 79, 384)-Net in Base 8 — Constructive

(49, 79, 384)-net in base 8, using

1 times m-reduction [i] based on (49, 80, 384)-net in base 8, using
- trace code for nets [i] based on (9, 40, 192)-net in base 64, using
  - 2 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
    - base change [i] based on digital (3, 36, 192)-net over F₁₂₈, using
      - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(49, 79, 514)-Net over F₈ — Digital

Digital (49, 79, 514)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷⁹, 514, F₈, 30) (dual of [514, 435, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁷⁹, 515, F₈, 30) (dual of [515, 436, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(28) [i] based on
    1. linear OA(8⁷⁹, 512, F₈, 30) (dual of [512, 433, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    2. linear OA(8⁷⁶, 512, F₈, 29) (dual of [512, 436, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(8⁰, 3, F₈, 0) (dual of [3, 3, 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]

(49, 79, 52343)-Net in Base 8 — Upper bound on s

There is no (49, 79, 52344)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 220857 575738 004099 304980 401381 612585 365130 337254 725985 794974 709084 935930 > 8⁷⁹ [i]