MinT - Best Known (187−29, 187, s)-Nets in Base 4

Best Known (187−29, 187, s)-Nets in Base 4

(187−29, 187, 4696)-Net over F₄ — Constructive and digital

Digital (158, 187, 4696)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 18, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, 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]

(187−29, 187, 51124)-Net over F₄ — Digital

Digital (158, 187, 51124)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁸⁷, 51124, F₄, 29) (dual of [51124, 50937, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁸⁷, 65558, F₄, 29) (dual of [65558, 65371, 30]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(4¹⁸, 22, F₄, 14) (dual of [22, 4, 15]-code), using
      - 1 times truncation [i] based on linear OA(4¹⁹, 23, F₄, 15) (dual of [23, 4, 16]-code), using
        constructionÂ X applied to C₁ âŠ‚ C₂ with C₁ a [17,1,16]-code [i] based on
        linear OA(4¹³, 17, F₄, 11) (dual of [17, 4, 12]-code), using projective code from ovoid in PG(3,Â 4) [i]
        linear OA(4³, 6, F₄, 3) (dual of [6, 3, 4]-code or 6-arc in PG(2,4) or 6-cap in PG(2,4)), using
        hyperoval in PG(2,Â 4) [i]
        the Hexacode [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]

(187−29, 187, large)-Net in Base 4 — Upper bound on s

There is no (158, 187, large)-net in base 4, because

27 times m-reduction [i] would yield (158, 160, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]