MinT - Best Known (259−32, 259, s)-Nets in Base 4

Best Known (259−32, 259, s)-Nets in Base 4

(259−32, 259, 65550)-Net over F₄ — Constructive and digital

Digital (227, 259, 65550)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 19, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (208, 240, 65536)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4²⁴⁰, 65536, F₄, 32, 32) (dual of [(65536, 32), 2096912, 33]-NRT-code), using
    - OA 16-folding and stacking [i] based on linear OA(4²⁴⁰, 1048576, F₄, 32) (dual of [1048576, 1048336, 33]-code), using
      - 1 times truncation [i] based on linear OA(4²⁴¹, 1048577, F₄, 33) (dual of [1048577, 1048336, 34]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]

(259−32, 259, 604458)-Net over F₄ — Digital

Digital (227, 259, 604458)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4²⁵⁹, 604458, F₄, 32) (dual of [604458, 604199, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(4²⁵⁹, 1048655, F₄, 32) (dual of [1048655, 1048396, 33]-code), using
  - strength reduction [i] based on linear OA(4²⁵⁹, 1048655, F₄, 33) (dual of [1048655, 1048396, 34]-code), using
    - constructionÂ X applied to C([0,16]) âŠ‚ C([0,12]) [i] based on
      1. linear OA(4²⁴¹, 1048577, F₄, 33) (dual of [1048577, 1048336, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,16], and minimum distance dÂ â‰¥ |{−16,−15,…,16}|+1Â =Â 34 (BCH-bound) [i]
      2. linear OA(4¹⁸¹, 1048577, F₄, 25) (dual of [1048577, 1048396, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 1048577Â | 4²⁰âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      3. linear OA(4¹⁸, 78, F₄, 7) (dual of [78, 60, 8]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹⁸, 85, F₄, 7) (dual of [85, 67, 8]-code), using
        the BCH-code C(I) with length 85Â | 4⁴âˆ’1, defining interval IÂ =Â {4,10,16,…,40}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 8 [i]

(259−32, 259, large)-Net in Base 4 — Upper bound on s

There is no (227, 259, large)-net in base 4, because

30 times m-reduction [i] would yield (227, 229, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]