MinT - Best Known (46, 74, s)-Nets in Base 8

Best Known (46, 74, s)-Nets in Base 8

(46, 74, 354)-Net over F₈ — Constructive and digital

Digital (46, 74, 354)-net over F₈, using

4 times m-reduction [i] based on digital (46, 78, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 39, 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]

(46, 74, 384)-Net in Base 8 — Constructive

(46, 74, 384)-net in base 8, using

trace code for nets [i] based on (9, 37, 192)-net in base 64, using
- 5 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]

(46, 74, 503)-Net over F₈ — Digital

Digital (46, 74, 503)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁷⁴, 503, F₈, 28) (dual of [503, 429, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁷⁴, 519, F₈, 28) (dual of [519, 445, 29]-code), using
  - constructionÂ X applied to C_e(27) âŠ‚ C_e(25) [i] based on
    1. linear OA(8⁷³, 512, F₈, 28) (dual of [512, 439, 29]-code), using an extension C_e(27) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,27], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 28 [i]
    2. linear OA(8⁶⁷, 512, F₈, 26) (dual of [512, 445, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
    3. linear OA(8¹, 7, F₈, 1) (dual of [7, 6, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(8¹, s, F₈, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(46, 74, 51258)-Net in Base 8 — Upper bound on s

There is no (46, 74, 51259)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 741462 394323 631753 730645 708893 507582 988413 534705 291874 915310 188508 > 8⁷⁴ [i]