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

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

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

Digital (3, m, 64)-net over F₃₂ for arbitrarily large m, 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, 3+∞, 131)-Net over F₃₂ — Upper bound on s (digital)

There is no digital (3, m, 132)-net over F₃₂ for arbitrarily large m, because

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, 3+∞, 132)-Net in Base 32 — Upper bound on s

There is no (3, m, 133)-net in base 32 for arbitrarily large m, because

m-reduction [i] would yield (3, 131, 133)-net in base 32, but
- extracting embedded OOA [i] would yield OA(32¹³¹, 133, S₃₂, 128), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 23 920328 665319 054867 909425 788093 943381 456209 876083 329888 835036 868243 751788 655030 496164 120160 199620 164471 448192 017206 646201 063596 209604 856372 278912 979945 202323 302617 838309 945901 490499 445045 874005 114880 / 129 > 32¹³¹ [i]