MinT - Best Known (236−173, 236, s)-Nets in Base 4

Best Known (236−173, 236, s)-Nets in Base 4

Digital (63, 236, 66)-net over F₄, using

t-expansion [i] based on digital (49, 236, 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]

Digital (63, 236, 99)-net over F₄, using

t-expansion [i] based on digital (61, 236, 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]

There is no digital (63, 236, 318)-net over F₄, because

1 times m-reduction [i] would yield digital (63, 235, 318)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²³⁵, 318, F₄, 172) (dual of [318, 83, 173]-code), but
  - residual code [i] would yield OA(4⁶³, 145, S₄, 43), but
    - the linear programming bound shows that MÂ â‰¥ 410852 595144 601422 661736 194475 352548 234222 916103 520325 789662 026727 829058 017732 594422 665880 071589 138518 417655 328617 787562 131456 / 4712 214848 200787 196384 454868 078661 057064 361851 730871 257084 069591 637575 047639 691128 659913 > 4⁶³ [i]

There is no (63, 236, 416)-net in base 4, because

1 times m-reduction [i] would yield (63, 235, 416)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3181 661023 930669 334098 532706 578191 271325 343425 100719 985412 219839 028346 278442 425560 672197 844243 014064 481081 431103 554870 495469 476086 164045 672590 > 4²³⁵ [i]