MinT - Best Known (35−11, 35, s)-Nets in Base 16

Best Known (35−11, 35, s)-Nets in Base 16

(35−11, 35, 1045)-Net over F₁₆ — Constructive and digital

Digital (24, 35, 1045)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 3, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (5, 10, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 5, 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]
3. digital (11, 22, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 11, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(35−11, 35, 5079)-Net over F₁₆ — Digital

Digital (24, 35, 5079)-net over F₁₆, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16³⁵, 5079, F₁₆, 11) (dual of [5079, 5044, 12]-code), using
- 976 step VarÅ¡amovâ€“Edel lengthening with (r_i)Â = (2, 14 times 0, 1, 178 times 0, 1, 781 times 0) [i] based on linear OA(16³¹, 4099, F₁₆, 11) (dual of [4099, 4068, 12]-code), using
  - constructionÂ X applied to C_e(10) âŠ‚ C_e(9) [i] based on
    1. linear OA(16³¹, 4096, F₁₆, 11) (dual of [4096, 4065, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
    2. linear OA(16²⁸, 4096, F₁₆, 10) (dual of [4096, 4068, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 16³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [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]

(35−11, 35, large)-Net in Base 16 — Upper bound on s

There is no (24, 35, large)-net in base 16, because

9 times m-reduction [i] would yield (24, 26, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]