MinT - Best Known (63, 239, s)-Nets in Base 4

Best Known (63, 239, s)-Nets in Base 4

Digital (63, 239, 66)-net over F₄, using

t-expansion [i] based on digital (49, 239, 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 (63, 239, 99)-net over F₄, using

t-expansion [i] based on digital (61, 239, 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 (63, 239, 302)-net over F₄, because

extracting embedded orthogonal array [i] would yield linear OA(4²³⁹, 302, F₄, 176) (dual of [302, 63, 177]-code), but
- residual code [i] would yield OA(4⁶³, 125, S₄, 44), but
  - the linear programming bound shows that MÂ â‰¥ 175234 933305 074206 543921 418495 881277 780883 332813 429919 101006 437404 404381 504833 864636 079141 500337 375376 052360 191696 611094 696757 939320 665090 885499 385380 009810 778380 200900 617655 056099 806688 415743 521650 982924 518318 871643 642600 732269 143252 854005 385711 951077 298974 645803 648886 013429 809152 / 2025 517705 388844 186066 624345 865386 010728 459225 322617 301863 147239 853832 944815 211835 885283 966428 532227 973928 412742 757997 253523 507788 055914 588259 352119 823340 513122 117611 644548 283151 687312 224909 613727 897424 811994 051452 501775 130990 943683 258327 507029 > 4⁶³ [i]

There is no (63, 239, 414)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 797162 062204 790481 056729 452551 983112 546350 410208 874820 518357 828529 193675 718873 681929 566298 093068 735376 085217 407337 752677 641518 110545 730960 185438 > 4²³⁹ [i]