MinT - Best Known (214−24, 214, s)-Nets in Base 4

Best Known (214−24, 214, s)-Nets in Base 4

(214−24, 214, 349540)-Net over F₄ — Constructive and digital

Digital (190, 214, 349540)-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 (174, 198, 349525)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹⁹⁸, 349525, F₄, 24, 24) (dual of [(349525, 24), 8388402, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(4¹⁹⁸, 4194300, F₄, 24) (dual of [4194300, 4194102, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹⁹⁸, 4194304, F₄, 24) (dual of [4194304, 4194106, 25]-code), using
        1 times truncation [i] based on linear OA(4¹⁹⁹, 4194305, F₄, 25) (dual of [4194305, 4194106, 26]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 4194305Â | 4²²âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(214−24, 214, 2097193)-Net over F₄ — Digital

Digital (190, 214, 2097193)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(4²¹⁴, 2097193, F₄, 2, 24) (dual of [(2097193, 2), 4194172, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4²¹⁴, 4194386, F₄, 24) (dual of [4194386, 4194172, 25]-code), using
  - strength reduction [i] based on linear OA(4²¹⁴, 4194386, F₄, 25) (dual of [4194386, 4194172, 26]-code), using
    - constructionÂ X applied to C([0,12]) âŠ‚ C([0,8]) [i] based on
      1. linear OA(4¹⁹⁹, 4194305, F₄, 25) (dual of [4194305, 4194106, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305Â | 4²²âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      2. linear OA(4¹³³, 4194305, F₄, 17) (dual of [4194305, 4194172, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305Â | 4²²âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
      3. linear OA(4¹⁵, 81, F₄, 7) (dual of [81, 66, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹⁵, 85, F₄, 7) (dual of [85, 70, 8]-code), using
        a code from Grasslâ€™s database [i]

(214−24, 214, large)-Net in Base 4 — Upper bound on s

There is no (190, 214, large)-net in base 4, because

22 times m-reduction [i] would yield (190, 192, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]