MinT - Best Known (140, 140+31, s)-Nets in Base 4

Best Known (140, 140+31, s)-Nets in Base 4

(140, 140+31, 1223)-Net over F₄ — Constructive and digital

Digital (140, 171, 1223)-net over F₄, using

4² times duplication [i] based on digital (138, 169, 1223)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (30, 45, 195)-net over F₄, using
    - trace code for nets [i] based on digital (0, 15, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. digital (93, 124, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 31, 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]

(140, 140+31, 13139)-Net over F₄ — Digital

Digital (140, 171, 13139)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹⁷¹, 13139, F₄, 31) (dual of [13139, 12968, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹⁷¹, 16421, F₄, 31) (dual of [16421, 16250, 32]-code), using
  - 2 times code embedding in larger space [i] based on linear OA(4¹⁶⁹, 16419, F₄, 31) (dual of [16419, 16250, 32]-code), using
    - constructionÂ X applied to C_e(30) âŠ‚ C_e(25) [i] based on
      1. linear OA(4¹⁶², 16384, F₄, 31) (dual of [16384, 16222, 32]-code), using an extension C_e(30) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
      2. linear OA(4¹³⁴, 16384, F₄, 26) (dual of [16384, 16250, 27]-code), using an extension C_e(25) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,25], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 26 [i]
      3. linear OA(4⁷, 35, F₄, 4) (dual of [35, 28, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(4⁷, 43, F₄, 4) (dual of [43, 36, 5]-code), using
        quaternary constacyclic codes with minimum distance 5 [i]

(140, 140+31, large)-Net in Base 4 — Upper bound on s

There is no (140, 171, large)-net in base 4, because

29 times m-reduction [i] would yield (140, 142, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]