MinT - Best Known (189, 230, s)-Nets in Base 4

Best Known (189, 230, s)-Nets in Base 4

(189, 230, 1553)-Net over F₄ — Constructive and digital

Digital (189, 230, 1553)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 23, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (166, 207, 1539)-net over F₄, using
  - trace code for nets [i] based on digital (28, 69, 513)-net over F₆₄, using
    - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(189, 230, 16450)-Net over F₄ — Digital

Digital (189, 230, 16450)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²³⁰, 16450, F₄, 41) (dual of [16450, 16220, 42]-code), using
- constructionÂ X applied to C_e(40) âŠ‚ C_e(30) [i] based on
  1. linear OA(4²¹¹, 16384, F₄, 41) (dual of [16384, 16173, 42]-code), using an extension C_e(40) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,40], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 41 [i]
  2. linear OA(4¹⁶², 16384, F₄, 31) (dual of [16384, 16222, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
  3. linear OA(4¹⁹, 66, F₄, 9) (dual of [66, 47, 10]-code), using
    - a code from Grasslâ€™s database [i]

(189, 230, large)-Net in Base 4 — Upper bound on s

There is no (189, 230, large)-net in base 4, because

39 times m-reduction [i] would yield (189, 191, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]