MinT - Best Known (120−25, 120, s)-Nets in Base 4

Best Known (120−25, 120, s)-Nets in Base 4

(120−25, 120, 1049)-Net over F₄ — Constructive and digital

Digital (95, 120, 1049)-net over F₄, using

4¹ times duplication [i] based on digital (94, 119, 1049)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 19, 21)-net over F₄, using
    - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (75, 100, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 25, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(120−25, 120, 4078)-Net over F₄ — Digital

Digital (95, 120, 4078)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²⁰, 4078, F₄, 25) (dual of [4078, 3958, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹²⁰, 4132, F₄, 25) (dual of [4132, 4012, 26]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(4¹¹⁸, 4130, F₄, 25) (dual of [4130, 4012, 26]-code), using
    - constructionÂ X applied to C([0,12]) âŠ‚ C([0,9]) [i] based on
      1. linear OA(4¹⁰⁹, 4097, F₄, 25) (dual of [4097, 3988, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]
      2. linear OA(4⁸⁵, 4097, F₄, 19) (dual of [4097, 4012, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      3. linear OA(4⁹, 33, F₄, 5) (dual of [33, 24, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁹, 36, F₄, 5) (dual of [36, 27, 6]-code), using
        codes constructed by inverting constructionÂ Y₁ [i]

(120−25, 120, 1646894)-Net in Base 4 — Upper bound on s

There is no (95, 120, 1646895)-net in base 4, because

1 times m-reduction [i] would yield (95, 119, 1646895)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 441711 887272 745008 111601 633446 355511 971388 547440 103561 140362 878477 075975 > 4¹¹⁹ [i]