MinT - Best Known (226−160, 226, s)-Nets in Base 4

Best Known (226−160, 226, s)-Nets in Base 4

(226−160, 226, 66)-Net over F₄ — Constructive and digital

Digital (66, 226, 66)-net over F₄, using

t-expansion [i] based on digital (49, 226, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(226−160, 226, 99)-Net over F₄ — Digital

Digital (66, 226, 99)-net over F₄, using

t-expansion [i] based on digital (61, 226, 99)-net over F₄, using
- net from sequence [i] based on digital (61, 98)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 61 and N(F) ≥ 99, using
    - Drinfeld modules of rank 1 [i]

(226−160, 226, 421)-Net over F₄ — Upper bound on s (digital)

There is no digital (66, 226, 422)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²²⁶, 422, F₄, 160) (dual of [422, 196, 161]-code), but
- residual code [i] would yield OA(4⁶⁶, 261, S₄, 40), but
  - the linear programming bound shows that MÂ â‰¥ 327016 444997 139005 823489 073156 791701 824991 264795 866993 154478 474670 227349 271740 416000 / 57 698874 415078 925905 565190 919370 722090 693979 > 4⁶⁶ [i]

(226−160, 226, 449)-Net in Base 4 — Upper bound on s

There is no (66, 226, 450)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13558 346418 288710 460236 276869 284953 762292 072260 121787 210130 665883 003269 982015 726439 119417 506931 895077 355080 771232 048534 466440 573153 257560 > 4²²⁶ [i]