MinT - Best Known (3, 105, s)-Nets in Base 32

Best Known (3, 105, s)-Nets in Base 32

(3, 105, 64)-Net over F₃₂ — Constructive and digital

Digital (3, 105, 64)-net over F₃₂, using

net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
  - a function field by SÃ©mirat [i]

(3, 105, 131)-Net over F₃₂ — Upper bound on s (digital)

There is no digital (3, 105, 132)-net over F₃₂, because

6 times m-reduction [i] would yield digital (3, 99, 132)-net over F₃₂, but
- extracting embedded orthogonal array [i] would yield linear OA(32⁹⁹, 132, F₃₂, 96) (dual of [132, 33, 97]-code), but
  - residual code [i] would yield OA(32³, 35, S₃₂, 3), but
    - 1 times truncation [i] would yield OA(32², 34, S₃₂, 2), but
      - bound for OAs with strength kÂ =Â 2 [i]
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1055 > 32² [i]

(3, 105, 224)-Net in Base 32 — Upper bound on s

There is no (3, 105, 225)-net in base 32, because

5 times m-reduction [i] would yield (3, 100, 225)-net in base 32, but
- extracting embedded orthogonal array [i] would yield OA(32¹⁰⁰, 225, S₃₂, 97), but
  - the linear programming bound shows that MÂ â‰¥ 2 176621 357211 405922 874416 376296 464536 271980 950849 045345 539822 984706 307743 412210 042179 441204 168005 917650 563663 281948 862935 325493 948854 158937 218572 505987 463250 532393 307695 218688 / 638810 956324 083601 474063 > 32¹⁰⁰ [i]