MinT - Best Known (179−25, 179, s)-Nets in Base 4

Best Known (179−25, 179, s)-Nets in Base 4

(179−25, 179, 21860)-Net over F₄ — Constructive and digital

Digital (154, 179, 21860)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 16, 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 (138, 163, 21845)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁶³, 21845, F₄, 25, 25) (dual of [(21845, 25), 545962, 26]-NRT-code), using
    - OOA 12-folding and stacking with additional row [i] based on linear OA(4¹⁶³, 262141, F₄, 25) (dual of [262141, 261978, 26]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁶³, 262144, F₄, 25) (dual of [262144, 261981, 26]-code), using
        an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 262143Â = 4⁹âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]

(179−25, 179, 143450)-Net over F₄ — Digital

Digital (154, 179, 143450)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷⁹, 143450, F₄, 25) (dual of [143450, 143271, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁷⁹, 262165, F₄, 25) (dual of [262165, 261986, 26]-code), using
  - (u,Â u+v)-construction [i] based on
    1. linear OA(4¹⁶, 20, F₄, 12) (dual of [20, 4, 13]-code), using
      - 3 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(4¹⁶³, 262145, F₄, 25) (dual of [262145, 261982, 26]-code), using
      - the expurgated narrow-sense BCH-code C(I) with length 262145Â | 4¹⁸âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(179−25, 179, large)-Net in Base 4 — Upper bound on s

There is no (154, 179, large)-net in base 4, because

23 times m-reduction [i] would yield (154, 156, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]