MinT - Best Known (123−70, 123, s)-Nets in Base 3

Best Known (123−70, 123, s)-Nets in Base 3

(123−70, 123, 48)-Net over F₃ — Constructive and digital

Digital (53, 123, 48)-net over F₃, using

t-expansion [i] based on digital (45, 123, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(123−70, 123, 64)-Net over F₃ — Digital

Digital (53, 123, 64)-net over F₃, using

t-expansion [i] based on digital (49, 123, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(123−70, 123, 297)-Net in Base 3 — Upper bound on s

There is no (53, 123, 298)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 52414 111926 749491 232389 819104 008631 713675 652417 722198 092785 > 3¹²³ [i]
extracting embedded orthogonal array [i] would yield OA(3¹²³, 298, S₃, 70), but
- 2 times code embedding in larger space [i] would yield OA(3¹²⁵, 300, S₃, 70), but
  - the linear programming bound shows that MÂ â‰¥ 2 916272 042802 359173 552813 695146 181366 464103 359775 885109 022526 939215 028038 650425 977694 498077 304526 540151 478913 893863 615320 348233 737890 298484 549087 155229 131244 245612 065191 077706 956469 190179 005571 235548 210866 193544 538862 869598 457475 427300 998737 040368 466999 759155 758135 325253 315591 256245 191539 272534 267166 964832 663784 188056 066126 619700 900113 611363 663961 265696 458300 092019 279727 976287 325797 653841 578984 759167 118660 865331 994018 757565 865324 292860 834832 037478 701835 855191 591913 512252 507830 706252 507573 071283 583152 689873 422759 168273 427031 666319 641213 187322 472266 360956 229478 009382 674121 592364 103751 266461 246651 971109 262692 647259 469098 434209 459939 137475 408192 156508 125822 122278 987363 104489 917387 483581 707713 578093 867161 192955 645586 827959 712633 / 4 556019 260270 167642 532166 262062 210308 185340 969664 425109 661093 742866 562915 414186 053935 151760 026309 941854 729652 314441 307528 803725 258322 790147 998866 918007 941615 235809 611573 029347 681539 842081 268112 349536 791529 585891 948769 219289 500254 499927 416013 352958 031686 525431 895436 651531 847018 321911 189322 830517 771008 549350 657874 290228 301043 987530 185805 264097 470053 485045 774420 588579 110101 232380 197268 966469 650365 988888 468058 971317 577810 147071 774906 454407 337793 195790 652413 848821 647188 903408 650720 647106 531625 119904 072009 967142 392607 639452 684840 319091 298955 320705 795140 701378 703744 202761 947188 625967 434743 658137 173068 943433 869279 619317 471020 734894 551771 684518 570037 320813 948285 624883 663797 > 3¹²⁵ [i]

Best Known (123−70, 123, s)-Nets in Base 3

(123−70, 123, 48)-Net over F3 — Constructive and digital

(123−70, 123, 64)-Net over F3 — Digital

(123−70, 123, 297)-Net in Base 3 — Upper bound on s

(123−70, 123, 48)-Net over F₃ — Constructive and digital

(123−70, 123, 64)-Net over F₃ — Digital