MinT - Best Known (59−22, 59, s)-Nets in Base 8

Best Known (59−22, 59, s)-Nets in Base 8

(59−22, 59, 354)-Net over F₈ — Constructive and digital

Digital (37, 59, 354)-net over F₈, using

1 times m-reduction [i] based on digital (37, 60, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 30, 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]

(59−22, 59, 482)-Net over F₈ — Digital

Digital (37, 59, 482)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁵⁹, 482, F₈, 22) (dual of [482, 423, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8⁵⁹, 519, F₈, 22) (dual of [519, 460, 23]-code), using
  - constructionÂ X applied to C_e(21) âŠ‚ C_e(19) [i] based on
    1. linear OA(8⁵⁸, 512, F₈, 22) (dual of [512, 454, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
    2. linear OA(8⁵², 512, F₈, 20) (dual of [512, 460, 21]-code), using an extension C_e(19) of the primitive narrow-sense BCH-code C(I) with length 511Â = 8³âˆ’1, defining interval IÂ =Â [1,19], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 20 [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]

(59−22, 59, 48944)-Net in Base 8 — Upper bound on s

There is no (37, 59, 48945)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 191582 445114 106362 665381 496804 974543 914946 755789 413712 > 8⁵⁹ [i]