MinT - Best Known (103−34, 103, s)-Nets in Base 16

Best Known (103−34, 103, s)-Nets in Base 16

(103−34, 103, 1028)-Net over F₁₆ — Constructive and digital

Digital (69, 103, 1028)-net over F₁₆, using

1 times m-reduction [i] based on digital (69, 104, 1028)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (17, 34, 514)-net over F₁₆, 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]
  2. digital (35, 70, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 35, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(103−34, 103, 5052)-Net over F₁₆ — Digital

Digital (69, 103, 5052)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹⁰³, 5052, F₁₆, 34) (dual of [5052, 4949, 35]-code), using
- 944 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (3, 0, 0, 1, 11 times 0, 1, 30 times 0, 1, 76 times 0, 1, 161 times 0, 1, 280 times 0, 1, 377 times 0) [i] based on linear OA(16⁹⁴, 4099, F₁₆, 34) (dual of [4099, 4005, 35]-code), using
  - constructionÂ X applied to C_e(33) âŠ‚ C_e(32) [i] based on
    1. linear OA(16⁹⁴, 4096, F₁₆, 34) (dual of [4096, 4002, 35]-code), using an extension C_e(33) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,33], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
    2. linear OA(16⁹¹, 4096, F₁₆, 33) (dual of [4096, 4005, 34]-code), using an extension C_e(32) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,32], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 33 [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]

(103−34, 103, large)-Net in Base 16 — Upper bound on s

There is no (69, 103, large)-net in base 16, because

32 times m-reduction [i] would yield (69, 71, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]