MinT - Best Known (189−29, 189, s)-Nets in Base 4

Best Known (189−29, 189, s)-Nets in Base 4

(189−29, 189, 4698)-Net over F₄ — Constructive and digital

Digital (160, 189, 4698)-net over F₄, using

4¹ times duplication [i] based on digital (159, 188, 4698)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 19, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (140, 169, 4681)-net over F₄, using
    - net defined by OOA [i] based on linear OOA(4¹⁶⁹, 4681, F₄, 29, 29) (dual of [(4681, 29), 135580, 30]-NRT-code), using
      - OOA 14-folding and stacking with additional row [i] based on linear OA(4¹⁶⁹, 65535, F₄, 29) (dual of [65535, 65366, 30]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹⁶⁹, 65536, F₄, 29) (dual of [65536, 65367, 30]-code), using
        an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(189−29, 189, 56656)-Net over F₄ — Digital

Digital (160, 189, 56656)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁹, 56656, F₄, 29) (dual of [56656, 56467, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁸⁹, 65562, F₄, 29) (dual of [65562, 65373, 30]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(4²⁰, 26, F₄, 14) (dual of [26, 6, 15]-code), using
      - discarding factors / shortening the dual code based on linear OA(4²⁰, 27, F₄, 14) (dual of [27, 7, 15]-code), using
        1 times truncation [i] based on linear OA(4²¹, 28, F₄, 15) (dual of [28, 7, 16]-code), using
        a quasi-cyclic code by Gulliver and Bhargava [i]
    2. linear OA(4¹⁶⁹, 65536, F₄, 29) (dual of [65536, 65367, 30]-code), using
      - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 65535Â = 4⁸âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(189−29, 189, large)-Net in Base 4 — Upper bound on s

There is no (160, 189, large)-net in base 4, because

27 times m-reduction [i] would yield (160, 162, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]