MinT - Best Known (25, 25+11, s)-Nets in Base 16

Best Known (25, 25+11, s)-Nets in Base 16

(25, 25+11, 1285)-Net over F₁₆ — Constructive and digital

Digital (25, 36, 1285)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (1, 4, 257)-net over F₁₆, using
  - net defined by OOA [i] based on linear OOA(16⁴, 257, F₁₆, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
    - appending kth column [i] based on linear OOA(16⁴, 257, F₁₆, 2, 3) (dual of [(257, 2), 510, 4]-NRT-code), using
      - OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(16⁴, 257, F₁₆, 3) (dual of [257, 253, 4]-code or 257-cap in PG(3,16)), using
        ovoid in PG(3,Â 16) [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)

(25, 25+11, 6532)-Net over F₁₆ — Digital

Digital (25, 36, 6532)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(25, 25+11, large)-Net in Base 16 — Upper bound on s

There is no (25, 36, large)-net in base 16, because

9 times m-reduction [i] would yield (25, 27, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]