MinT - Best Known (31, 222, s)-Nets in Base 4

Best Known (31, 222, s)-Nets in Base 4

Digital (31, 222, 34)-net over F₄, using

t-expansion [i] based on digital (21, 222, 34)-net over F₄, using
- net from sequence [i] based on digital (21, 33)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
    - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(31, 222, 43)-net in base 4, using

t-expansion [i] based on (30, 222, 43)-net in base 4, using
- net from sequence [i] based on (30, 42)-sequence in base 4, using
  - base expansion [i] based on digital (60, 42)-sequence over F₂, using
    - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

Digital (31, 222, 60)-net over F₄, using

There is no (31, 222, 112)-net in base 4, because

2 times m-reduction [i] would yield (31, 220, 112)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²²⁰, 112, S₄, 2, 189), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 363 419362 147803 445274 661903 944002 267176 820680 343659 030140 745099 590319 644056 698961 663095 525356 881782 780381 260803 133088 966767 300814 307328 / 95 > 4²²⁰ [i]