MinT - Best Known (252−220, 252, s)-Nets in Base 4

Best Known (252−220, 252, s)-Nets in Base 4

Digital (32, 252, 34)-net over F₄, using

t-expansion [i] based on digital (21, 252, 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]

(32, 252, 43)-net in base 4, using

t-expansion [i] based on (30, 252, 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 (32, 252, 60)-net over F₄, using

t-expansion [i] based on digital (31, 252, 60)-net over F₄, using
- net from sequence [i] based on digital (31, 59)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 31 and N(F) ≥ 60, using
    - Drinfeld modules of rank 1 [i]

There is no (32, 252, 115)-net in base 4, because

26 times m-reduction [i] would yield (32, 226, 115)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4²²⁶, 115, S₄, 2, 194), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 186070 713419 675363 980626 894819 329160 794532 188335 953423 432061 490990 243657 757029 868371 504908 982723 472783 555205 531204 141550 984858 016925 351936 / 13 > 4²²⁶ [i]