MinT - Best Known (65, 65+179, s)-Nets in Base 4

Best Known (65, 65+179, s)-Nets in Base 4

Digital (65, 244, 66)-net over F₄, using

t-expansion [i] based on digital (49, 244, 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 (65, 244, 99)-net over F₄, using

t-expansion [i] based on digital (61, 244, 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 (65, 244, 333)-net over F₄, because

3 times m-reduction [i] would yield digital (65, 241, 333)-net over F₄, but
- extracting embedded orthogonal array [i] would yield linear OA(4²⁴¹, 333, F₄, 176) (dual of [333, 92, 177]-code), but
  - residual code [i] would yield OA(4⁶⁵, 156, S₄, 44), but
    - the linear programming bound shows that MÂ â‰¥ 5305 672794 944537 381320 515598 514645 629008 946704 424695 546266 632663 854579 036277 493526 466626 448910 185539 633024 632652 385646 851129 344000 / 3 530621 465654 281569 250779 664225 038700 823798 252333 134165 234221 216150 895884 348866 248497 833837 > 4⁶⁵ [i]

There is no (65, 244, 429)-net in base 4, because

1 times m-reduction [i] would yield (65, 243, 429)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 230 415694 967438 405261 582905 463454 376225 878522 402110 467129 054537 933695 069378 922828 736939 748426 916978 376016 869683 220965 919371 805678 508433 523429 382400 > 4²⁴³ [i]