MinT - Best Known (28, 39, s)-Nets in Base 16

Best Known (28, 39, s)-Nets in Base 16

(28, 39, 1559)-Net over F₁₆ — Constructive and digital

Digital (28, 39, 1559)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 2, 17)-net over F₁₆, using
  - kÂ =Â 2 construction [i]
2. digital (2, 5, 514)-net over F₁₆, using
  - s-reduction based on digital (2, 5, 629)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16⁵, 629, F₁₆, 3, 3) (dual of [(629, 3), 1882, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(16⁵, 629, F₁₆, 2, 3) (dual of [(629, 2), 1253, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(16⁵, 629, F₁₆, 3) (dual of [629, 624, 4]-code or 629-cap in PG(4,16)), using
        caps completed using a computer search [i]
3. 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]
4. 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)

(28, 39, 6554)-Net in Base 16 — Constructive

(28, 39, 6554)-net in base 16, using

net defined by OOA [i] based on OOA(16³⁹, 6554, S₁₆, 11, 11), using
- OOA 5-folding and stacking with additional row [i] based on OA(16³⁹, 32771, S₁₆, 11), using
  - discarding parts of the base [i] based on linear OA(32³¹, 32771, F₃₂, 11) (dual of [32771, 32740, 12]-code), using
    - constructionÂ X applied to C_e(10) âŠ‚ C_e(9) [i] based on
      1. linear OA(32³¹, 32768, F₃₂, 11) (dual of [32768, 32737, 12]-code), using an extension C_e(10) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 11 [i]
      2. linear OA(32²⁸, 32768, F₃₂, 10) (dual of [32768, 32740, 11]-code), using an extension C_e(9) of the primitive narrow-sense BCH-code C(I) with length 32767Â = 32³âˆ’1, defining interval IÂ =Â [1,9], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 10 [i]
      3. linear OA(32⁰, 3, F₃₂, 0) (dual of [3, 3, 1]-code), using
        discarding factors / shortening the dual code based on linear OA(32⁰, s, F₃₂, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
        OA with only one run / dual of code without redundancy [i]

(28, 39, 15000)-Net over F₁₆ — Digital

Digital (28, 39, 15000)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(28, 39, large)-Net in Base 16 — Upper bound on s

There is no (28, 39, large)-net in base 16, because

9 times m-reduction [i] would yield (28, 30, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]