MinT - Best Known (83, 126, s)-Nets in Base 16

Best Known (83, 126, s)-Nets in Base 16

(83, 126, 617)-Net over F₁₆ — Constructive and digital

Digital (83, 126, 617)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (19, 40, 103)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (3, 13, 38)-net over F₁₆, using
      - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
    2. digital (6, 27, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]
2. digital (43, 86, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 43, 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]

(83, 126, 664)-Net in Base 16 — Constructive

(83, 126, 664)-net in base 16, using

16¹ times duplication [i] based on (82, 125, 664)-net in base 16, using
- (u,Â u+v)-construction [i] based on
  1. (18, 39, 150)-net in base 16, using
    - base change [i] based on (5, 26, 150)-net in base 64, using
      - 2 times m-reduction [i] based on (5, 28, 150)-net in base 64, using
        base change [i] based on digital (1, 24, 150)-net over F₁₂₈, using
        net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
  2. digital (43, 86, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 43, 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]

(83, 126, 4538)-Net over F₁₆ — Digital

Digital (83, 126, 4538)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²⁶, 4538, F₁₆, 43) (dual of [4538, 4412, 44]-code), using
- 434 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 8 times 0, 1, 45 times 0, 1, 135 times 0, 1, 242 times 0) [i] based on linear OA(16¹²¹, 4099, F₁₆, 43) (dual of [4099, 3978, 44]-code), using
  - constructionÂ X applied to C_e(42) âŠ‚ C_e(41) [i] based on
    1. linear OA(16¹²¹, 4096, F₁₆, 43) (dual of [4096, 3975, 44]-code), using an extension C_e(42) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,42], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 43 [i]
    2. linear OA(16¹¹⁸, 4096, F₁₆, 42) (dual of [4096, 3978, 43]-code), using an extension C_e(41) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,41], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 42 [i]
    3. linear OA(16⁰, 3, F₁₆, 0) (dual of [3, 3, 1]-code), using
      - discarding factors / shortening the dual code based on linear OA(16⁰, s, F₁₆, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(83, 126, large)-Net in Base 16 — Upper bound on s

There is no (83, 126, large)-net in base 16, because

41 times m-reduction [i] would yield (83, 85, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]