MinT - Best Known (109, 138, s)-Nets in Base 4

Best Known (109, 138, s)-Nets in Base 4

(109, 138, 1049)-Net over F₄ — Constructive and digital

Digital (109, 138, 1049)-net over F₄, using

4¹ times duplication [i] based on digital (108, 137, 1049)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 21, 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 (87, 116, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 29, 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]

(109, 138, 4111)-Net over F₄ — Digital

Digital (109, 138, 4111)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³⁸, 4111, F₄, 29) (dual of [4111, 3973, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹³⁸, 4126, F₄, 29) (dual of [4126, 3988, 30]-code), using
  - constructionÂ X applied to C([0,14]) âŠ‚ C([0,12]) [i] based on
    1. linear OA(4¹³³, 4097, F₄, 29) (dual of [4097, 3964, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,14], and minimum distance dÂ â‰¥ |{−14,−13,…,14}|+1Â =Â 30 (BCH-bound) [i]
    2. 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]
    3. linear OA(4⁵, 29, F₄, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
      - discarding factors / shortening the dual code based on linear OA(4⁵, 41, F₄, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
        a cap found by Tallinii [i]

(109, 138, 1570075)-Net in Base 4 — Upper bound on s

There is no (109, 138, 1570076)-net in base 4, because

1 times m-reduction [i] would yield (109, 137, 1570076)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 30354 417814 382712 599781 480965 093478 690092 826424 643038 808079 882798 378452 559218 431036 > 4¹³⁷ [i]