MinT - Best Known (139−24, 139, s)-Nets in Base 4

Best Known (139−24, 139, s)-Nets in Base 4

(139−24, 139, 1374)-Net over F₄ — Constructive and digital

Digital (115, 139, 1374)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (102, 126, 1365)-net over F₄, using
  - net defined by OOA [i] based on linear OOA(4¹²⁶, 1365, F₄, 24, 24) (dual of [(1365, 24), 32634, 25]-NRT-code), using
    - OA 12-folding and stacking [i] based on linear OA(4¹²⁶, 16380, F₄, 24) (dual of [16380, 16254, 25]-code), using
      - discarding factors / shortening the dual code based on linear OA(4¹²⁶, 16384, F₄, 24) (dual of [16384, 16258, 25]-code), using
        1 times truncation [i] based on linear OA(4¹²⁷, 16385, F₄, 25) (dual of [16385, 16258, 26]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16385Â | 4¹⁴âˆ’1, defining interval IÂ =Â [0,12], and minimum distance dÂ â‰¥ |{−12,−11,…,12}|+1Â =Â 26 (BCH-bound) [i]

(139−24, 139, 1415)-Net in Base 4 — Constructive

(115, 139, 1415)-net in base 4, using

4¹ times duplication [i] based on (114, 138, 1415)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (30, 42, 387)-net in base 4, using
    - trace code for nets [i] based on (2, 14, 129)-net in base 64, using
      - base change [i] based on digital (0, 12, 129)-net over F₁₂₈, using
        net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
        Niederreiter sequence [i]
  2. digital (72, 96, 1028)-net over F₄, using
    - trace code for nets [i] based on digital (0, 24, 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]

(139−24, 139, 16433)-Net over F₄ — Digital

Digital (115, 139, 16433)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹³⁹, 16433, F₄, 24) (dual of [16433, 16294, 25]-code), using
- constructionÂ X with VarÅ¡amov bound [i] based on
  1. linear OA(4¹³⁷, 16429, F₄, 24) (dual of [16429, 16292, 25]-code), using
    - constructionÂ X applied to C_e(24) âŠ‚ C_e(17) [i] based on
      1. linear OA(4¹²⁷, 16384, F₄, 25) (dual of [16384, 16257, 26]-code), using an extension C_e(24) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,24], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 25 [i]
      2. linear OA(4⁹², 16384, F₄, 18) (dual of [16384, 16292, 19]-code), using an extension C_e(17) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,17], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 18 [i]
      3. linear OA(4¹⁰, 45, F₄, 5) (dual of [45, 35, 6]-code), using
        discarding factors / shortening the dual code based on linear OA(4¹⁰, 63, F₄, 5) (dual of [63, 53, 6]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 63Â = 4³âˆ’1, defining interval IÂ =Â [0,3], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
  2. linear OA(4¹³⁷, 16431, F₄, 23) (dual of [16431, 16294, 24]-code), using Gilbertâ€“VarÅ¡amov bound and b^mÂ = 4¹³⁷Â > V_b^sâˆ’1(kâˆ’1)Â = 15276 112369 899042 865413 274680 127703 757431 152895 921510 109551 812602 894225 601989 727384 [i]
  3. linear OA(4⁰, 2, F₄, 0) (dual of [2, 2, 1]-code), using
    - discarding factors / shortening the dual code based on linear OA(4⁰, s, F₄, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
      - OA with only one run / dual of code without redundancy [i]

(139−24, 139, large)-Net in Base 4 — Upper bound on s

There is no (115, 139, large)-net in base 4, because

22 times m-reduction [i] would yield (115, 117, large)-net in base 4, but
- mutually orthogonal hypercube bound [i]

Best Known (139−24, 139, s)-Nets in Base 4

(139−24, 139, 1374)-Net over F4 — Constructive and digital

(139−24, 139, 1415)-Net in Base 4 — Constructive

(139−24, 139, 16433)-Net over F4 — Digital

(139−24, 139, large)-Net in Base 4 — Upper bound on s

(139−24, 139, 1374)-Net over F₄ — Constructive and digital

(139−24, 139, 16433)-Net over F₄ — Digital