MinT - Best Known (86, 86+31, s)-Nets in Base 4

Best Known (86, 86+31, s)-Nets in Base 4

(86, 86+31, 531)-Net over F₄ — Constructive and digital

Digital (86, 117, 531)-net over F₄, using

t-expansion [i] based on digital (85, 117, 531)-net over F₄, 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]

(86, 86+31, 974)-Net over F₄ — Digital

Digital (86, 117, 974)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹¹⁷, 974, F₄, 31) (dual of [974, 857, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹¹⁷, 1035, F₄, 31) (dual of [1035, 918, 32]-code), using
  - constructionÂ X applied to C_e(30) âŠ‚ C_e(28) [i] based on
    1. 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]
    2. linear OA(4¹⁰⁶, 1024, F₄, 29) (dual of [1024, 918, 30]-code), using an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 1023Â = 4⁵âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
    3. linear OA(4¹, 11, F₄, 1) (dual of [11, 10, 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]

(86, 86+31, 96944)-Net in Base 4 — Upper bound on s

There is no (86, 117, 96945)-net in base 4, because

1 times m-reduction [i] would yield (86, 116, 96945)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6902 504685 941271 708386 552205 213985 890512 441326 860849 878089 633365 518512 > 4¹¹⁶ [i]