MinT - Best Known (109−36, 109, s)-Nets in Base 16

Best Known (109−36, 109, s)-Nets in Base 16

(109−36, 109, 1028)-Net over F₁₆ — Constructive and digital

Digital (73, 109, 1028)-net over F₁₆, using

1 times m-reduction [i] based on digital (73, 110, 1028)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (18, 36, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 18, 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]
  2. digital (37, 74, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 37, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(109−36, 109, 5232)-Net over F₁₆ — Digital

Digital (73, 109, 5232)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁰⁹, 5232, F₁₆, 36) (dual of [5232, 5123, 37]-code), using
- 1124 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 0, 1, 14 times 0, 1, 46 times 0, 1, 117 times 0, 1, 224 times 0, 1, 326 times 0, 1, 387 times 0) [i] based on linear OA(16¹⁰⁰, 4099, F₁₆, 36) (dual of [4099, 3999, 37]-code), using
  - constructionÂ X applied to C_e(35) âŠ‚ C_e(34) [i] based on
    1. linear OA(16¹⁰⁰, 4096, F₁₆, 36) (dual of [4096, 3996, 37]-code), using an extension C_e(35) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,35], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 36 [i]
    2. linear OA(16⁹⁷, 4096, F₁₆, 35) (dual of [4096, 3999, 36]-code), using an extension C_e(34) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,34], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 35 [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]

(109−36, 109, large)-Net in Base 16 — Upper bound on s

There is no (73, 109, large)-net in base 16, because

34 times m-reduction [i] would yield (73, 75, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]