MinT - Best Known (84, 84+30, s)-Nets in Base 4

Best Known (84, 84+30, s)-Nets in Base 4

(84, 84+30, 531)-Net over F₄ — Constructive and digital

Digital (84, 114, 531)-net over F₄, using

t-expansion [i] based on digital (83, 114, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 38, 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]

(84, 84+30, 992)-Net over F₄ — Digital

Digital (84, 114, 992)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁴, 992, F₄, 30) (dual of [992, 878, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹¹⁴, 1037, F₄, 30) (dual of [1037, 923, 31]-code), using
  - constructionÂ X applied to C_e(29) âŠ‚ C_e(26) [i] based on
    1. linear OA(4¹¹¹, 1024, F₄, 30) (dual of [1024, 913, 31]-code), using an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]
    2. linear OA(4¹⁰¹, 1024, F₄, 27) (dual of [1024, 923, 28]-code), using an extension C_e(26) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,26], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 27 [i]
    3. linear OA(4³, 13, F₄, 2) (dual of [13, 10, 3]-code), using
      - discarding factors / shortening the dual code based on linear OA(4³, 21, F₄, 2) (dual of [21, 18, 3]-code), using
        Hamming code H(3,4) [i]

(84, 84+30, 80581)-Net in Base 4 — Upper bound on s

There is no (84, 114, 80582)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 431 379813 477175 770085 860026 971048 881039 051615 232129 112139 409417 427488 > 4¹¹⁴ [i]