MinT - Best Known (89, 89+33, s)-Nets in Base 4

Best Known (89, 89+33, s)-Nets in Base 4

(89, 89+33, 531)-Net over F₄ — Constructive and digital

Digital (89, 122, 531)-net over F₄, using

1 times m-reduction [i] based on digital (89, 123, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 41, 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]

(89, 89+33, 904)-Net over F₄ — Digital

Digital (89, 122, 904)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²², 904, F₄, 33) (dual of [904, 782, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹²², 1030, F₄, 33) (dual of [1030, 908, 34]-code), using
  - constructionÂ X applied to C_e(32) âŠ‚ C_e(30) [i] based on
    1. linear OA(4¹²¹, 1024, F₄, 33) (dual of [1024, 903, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [i]
    2. linear OA(4¹¹⁶, 1024, F₄, 31) (dual of [1024, 908, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
    3. linear OA(4¹, 6, F₄, 1) (dual of [6, 5, 2]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹, s, F₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(89, 89+33, 80989)-Net in Base 4 — Upper bound on s

There is no (89, 122, 80990)-net in base 4, because

1 times m-reduction [i] would yield (89, 121, 80990)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 068443 220599 892192 244021 249513 213904 947773 986947 600193 104067 353391 049535 > 4¹²¹ [i]