MinT - Best Known (248−182, 248, s)-Nets in Base 4

Best Known (248−182, 248, s)-Nets in Base 4

Digital (66, 248, 66)-net over F₄, using

t-expansion [i] based on digital (49, 248, 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 (66, 248, 99)-net over F₄, using

t-expansion [i] based on digital (61, 248, 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 (66, 248, 331)-net over F₄, because

2 times m-reduction [i] would yield digital (66, 246, 331)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴⁶, 331, F₄, 180) (dual of [331, 85, 181]-code), but
  - residual code [i] would yield OA(4⁶⁶, 150, S₄, 45), but
    - the linear programming bound shows that MÂ â‰¥ 218 603248 395353 963143 573129 094845 574107 874696 792745 859935 933646 235876 315511 128217 770614 833515 398023 492482 446101 600214 474162 958366 887001 317163 943539 507200 000000 / 39636 522622 025040 367641 896273 764902 619875 534086 883796 962530 116095 725635 815042 105519 857817 694911 172894 869760 633227 396889 > 4⁶⁶ [i]

There is no (66, 248, 434)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 206332 984355 022589 829486 934855 775686 966927 477882 014679 345217 084535 088610 864387 349231 002182 314637 197892 772171 423135 655240 904102 664808 829204 434195 613280 > 4²⁴⁸ [i]