MinT - Best Known (81−17, 81, s)-Nets in Base 4

Best Known (81−17, 81, s)-Nets in Base 4

(81−17, 81, 1045)-Net over F₄ — Constructive and digital

Digital (64, 81, 1045)-net over F₄, using

4¹ times duplication [i] based on digital (63, 80, 1045)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (4, 12, 17)-net over F₄, using
    - linear code embedding found in a computer search [i]
  2. digital (51, 68, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 17, 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]

(81−17, 81, 3469)-Net over F₄ — Digital

Digital (64, 81, 3469)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4⁸¹, 3469, F₄, 17) (dual of [3469, 3388, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(4⁸¹, 4106, F₄, 17) (dual of [4106, 4025, 18]-code), using
  - constructionÂ X applied to C([0,8]) âŠ‚ C([1,8]) [i] based on
    1. linear OA(4⁷³, 4097, F₄, 17) (dual of [4097, 4024, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [0,8], and minimum distance dÂ â‰¥ |{−8,−7,…,8}|+1Â =Â 18 (BCH-bound) [i]
    2. linear OA(4⁷², 4097, F₄, 8) (dual of [4097, 4025, 9]-code), using the narrow-sense BCH-code C(I) with length 4097Â | 4¹²âˆ’1, defining interval IÂ =Â [1,8], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 9 [i]
    3. linear OA(4⁸, 9, F₄, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,4)), using
      - dual of repetition code with length 9 [i]

(81−17, 81, 1315729)-Net in Base 4 — Upper bound on s

There is no (64, 81, 1315730)-net in base 4, because

1 times m-reduction [i] would yield (64, 80, 1315730)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 461504 024015 958564 069072 621259 119339 823348 408134 > 4⁸⁰ [i]