MinT - Best Known (128, 128+∞, s)-Nets in Base 3

Best Known (128, 128+∞, s)-Nets in Base 3

Digital (128, m, 78)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (128, 77)-sequence over F₃, using
- t-expansion [i] based on digital (121, 77)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

Digital (128, m, 120)-net over F₃ for arbitrarily large m, using

There is no (128, m, 270)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (128, 1613, 270)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁶¹³, 270, S₃, 6, 1485), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 35193 427651 422670 085287 684352 848178 400093 983355 308709 644921 968854 651262 474281 277524 440810 269313 957812 066704 209456 429593 216182 315826 249983 289476 962567 194797 603372 977002 839750 915205 017318 741280 277908 752079 595652 464247 924540 470752 260706 288815 547706 426598 709307 936807 815434 917811 954503 118586 335786 954895 576046 063839 393450 819288 206740 772602 611658 654862 403488 499418 060521 874911 198301 307132 540222 155415 512519 839603 177629 695317 970811 345855 072421 884797 396122 332638 348023 423637 071823 377743 165741 727181 866693 468445 284200 083133 019172 568073 158346 688565 925803 618637 896644 130994 949203 379047 651090 707167 762916 708948 030052 256791 861585 677737 433359 209722 743586 726494 778592 444909 302111 888382 533175 235134 024524 508256 472598 957845 272855 938558 315686 319879 346456 085155 098325 692849 128394 869531 677793 / 743 > 3¹⁶¹³ [i]