MinT - Best Known (41−9, 41, s)-Nets in Base 4

Best Known (41−9, 41, s)-Nets in Base 4

(41−9, 41, 1038)-Net over F₄ — Constructive and digital

Digital (32, 41, 1038)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 5, 10)-net over F₄, using
  - linear code embedding found in a computer search [i]
2. digital (27, 36, 1028)-net over F₄, using
  - trace code for nets [i] based on digital (0, 9, 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]

(41−9, 41, 3101)-Net over F₄ — Digital

Digital (32, 41, 3101)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁴¹, 3101, F₄, 9) (dual of [3101, 3060, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁴¹, 4102, F₄, 9) (dual of [4102, 4061, 10]-code), using
  - constructionÂ X applied to C([0,4]) âŠ‚ C([1,4]) [i] based on
    1. linear OA(4³⁷, 4097, F₄, 9) (dual of [4097, 4060, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,4], and minimum distance dÂ â‰¥ |{−4,−3,…,4}|+1Â =Â 10 (BCH-bound) [i]
    2. linear OA(4³⁶, 4097, F₄, 4) (dual of [4097, 4061, 5]-code), using the narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [1,4], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]
    3. linear OA(4⁴, 5, F₄, 4) (dual of [5, 1, 5]-code or 5-arc in PG(3,4)), using
      - dual of repetition code with length 5 [i]

(41−9, 41, 773623)-Net in Base 4 — Upper bound on s

There is no (32, 41, 773624)-net in base 4, because

1 times m-reduction [i] would yield (32, 40, 773624)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 208926 442025 152279 588239 > 4⁴⁰ [i]