MinT - Best Known (257−231, 257, s)-Nets in Base 4

Best Known (257−231, 257, s)-Nets in Base 4

Digital (26, 257, 34)-net over F₄, using

t-expansion [i] based on digital (21, 257, 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]

(26, 257, 36)-net in base 4, using

net from sequence [i] based on (26, 35)-sequence in base 4, using
- base expansion [i] based on digital (52, 35)-sequence over F₂, using
  - t-expansion [i] based on digital (51, 35)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

Digital (26, 257, 55)-net over F₄, using

There is no (26, 257, 95)-net in base 4, because

71 times m-reduction [i] would yield (26, 186, 95)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(4¹⁸⁶, 95, S₄, 2, 160), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1 693054 953751 325278 652615 004303 045905 687020 046777 812161 513403 232213 683357 369020 768459 206753 139586 218940 909796 458496 / 161 > 4¹⁸⁶ [i]