MinT - Best Known (68, 68+192, s)-Nets in Base 4

Best Known (68, 68+192, s)-Nets in Base 4

Digital (68, 260, 66)-net over F₄, using

t-expansion [i] based on digital (49, 260, 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, 260, 99)-net over F₄, using

t-expansion [i] based on digital (61, 260, 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, 260, 310)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²⁶⁰, 310, F₄, 192) (dual of [310, 50, 193]-code), but
- residual code [i] would yield OA(4⁶⁸, 117, S₄, 48), but
  - the linear programming bound shows that MÂ â‰¥ 17 386566 400310 365130 429344 698706 546300 548809 167457 041405 229063 271574 299683 241127 057478 266601 477863 676931 347562 102784 / 194 255471 902233 779114 903901 006951 502367 530957 736648 427641 695122 360967 936465 > 4⁶⁸ [i]

There is no (68, 260, 445)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 632657 361639 771497 771047 811282 666008 933641 833883 244931 365302 111093 935754 039798 832066 112335 762649 761701 783776 128706 493214 652351 737025 709083 814370 977913 058410 > 4²⁶⁰ [i]