MinT - Best Known (68, 253, s)-Nets in Base 4

Best Known (68, 253, s)-Nets in Base 4

Digital (68, 253, 66)-net over F₄, using

t-expansion [i] based on digital (49, 253, 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 (68, 253, 99)-net over F₄, using

t-expansion [i] based on digital (61, 253, 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 (68, 253, 345)-net over F₄, because

1 times m-reduction [i] would yield digital (68, 252, 345)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁵², 345, F₄, 184) (dual of [345, 93, 185]-code), but
  - residual code [i] would yield OA(4⁶⁸, 160, S₄, 46), but
    - the linear programming bound shows that MÂ â‰¥ 10178 629433 929836 486543 068800 140163 269042 043780 022134 664338 507769 510819 622533 364638 543087 235674 689592 586232 641534 482804 572425 994198 736437 248000 / 113203 572801 045878 145860 058818 437218 882917 411800 440768 742206 357256 206420 022337 868833 760142 804084 942919 > 4⁶⁸ [i]

There is no (68, 253, 449)-net in base 4, because

1 times m-reduction [i] would yield (68, 252, 449)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 58 402667 923788 321011 448134 744652 035030 129169 335616 802847 454570 181078 351357 929061 423747 765944 208009 174616 202401 084378 428295 222101 055267 721389 973346 410400 > 4²⁵² [i]