MinT - Best Known (52−17, 52, s)-Nets in Base 4

Best Known (52−17, 52, s)-Nets in Base 4

(52−17, 52, 195)-Net over F₄ — Constructive and digital

Digital (35, 52, 195)-net over F₄, using

4¹ times duplication [i] based on digital (34, 51, 195)-net over F₄, using
- trace code for nets [i] based on digital (0, 17, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]

(52−17, 52, 228)-Net over F₄ — Digital

Digital (35, 52, 228)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁵², 228, F₄, 17) (dual of [228, 176, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁵², 263, F₄, 17) (dual of [263, 211, 18]-code), using
  - constructionÂ X applied to C([0,8]) âŠ‚ C([0,6]) [i] based on
    1. linear OA(4⁴⁹, 257, F₄, 17) (dual of [257, 208, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 257Â | 4⁸âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
    2. linear OA(4⁴¹, 257, F₄, 13) (dual of [257, 216, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 257Â | 4⁸âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
    3. 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]

(52−17, 52, 8637)-Net in Base 4 — Upper bound on s

There is no (35, 52, 8638)-net in base 4, because

1 times m-reduction [i] would yield (35, 51, 8638)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 072637 975299 121456 957074 824645 > 4⁵¹ [i]