MinT - Best Known (258−191, 258, s)-Nets in Base 4

Best Known (258−191, 258, s)-Nets in Base 4

Digital (67, 258, 66)-net over F₄, using

t-expansion [i] based on digital (49, 258, 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 (67, 258, 99)-net over F₄, using

t-expansion [i] based on digital (61, 258, 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 (67, 258, 312)-net over F₄, because

3 times m-reduction [i] would yield digital (67, 255, 312)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵⁵, 312, F₄, 188) (dual of [312, 57, 189]-code), but
  - residual code [i] would yield OA(4⁶⁷, 123, S₄, 47), but
    - the linear programming bound shows that MÂ â‰¥ 43402 647924 975488 920450 003944 566378 964570 407885 930315 158568 274900 052775 055729 425702 074252 778284 519466 230412 079171 949623 812197 935803 959581 270319 073009 003679 424611 406027 085968 310272 / 1 972722 979992 492582 380725 903751 953752 652629 461130 906381 054114 627794 331612 476321 726116 126868 678992 555081 191829 266311 527075 509736 859232 980625 > 4⁶⁷ [i]

There is no (67, 258, 439)-net in base 4, because

1 times m-reduction [i] would yield (67, 257, 439)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 64034 941302 874740 928383 291150 414340 113042 964608 913543 022061 315318 516420 337415 195552 627485 582715 139398 551661 787227 304677 716767 850882 639651 391039 749732 132000 > 4²⁵⁷ [i]