MinT - Best Known (172−102, 172, s)-Nets in Base 3

Best Known (172−102, 172, s)-Nets in Base 3

(172−102, 172, 48)-Net over F₃ — Constructive and digital

Digital (70, 172, 48)-net over F₃, using

t-expansion [i] based on digital (45, 172, 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]

(172−102, 172, 82)-Net over F₃ — Digital

Digital (70, 172, 82)-net over F₃, using

t-expansion [i] based on digital (69, 172, 82)-net over F₃, using
- net from sequence [i] based on digital (69, 81)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 69 and N(F) ≥ 82, using
    - Drinfeld modules of rank 1 [i]

(172−102, 172, 296)-Net in Base 3 — Upper bound on s

There is no (70, 172, 297)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3¹⁷², 297, S₃, 102), but
- 3 times code embedding in larger space [i] would yield OA(3¹⁷⁵, 300, S₃, 102), but
  - the linear programming bound shows that MÂ â‰¥ 16 198554 364679 432381 489686 101577 840909 983419 430915 343872 185033 381553 576946 076069 579907 001473 540702 214710 821998 887726 909125 829544 914665 448273 530074 943808 696693 037964 485084 157694 132521 254567 606707 219857 163357 246517 538784 808004 965389 309918 655222 313453 443943 109586 407376 415028 985580 321266 367340 164524 104226 989149 233336 390856 463359 722576 318481 785311 327280 451104 785757 592397 117700 879064 078249 264905 636905 877948 229404 150952 543963 589961 272276 182579 368658 022152 445883 363557 290483 270901 901200 238468 165725 131627 760319 527127 764114 602830 664643 879801 853958 021629 500328 511092 986298 330703 325794 796413 027915 391790 834072 969240 925465 855107 750448 623341 936469 862905 654153 127687 815436 138025 231153 / 37 720913 581798 855206 385351 973939 491254 808902 595946 642286 990299 637277 630436 143544 256842 437725 067503 499553 675300 109415 063335 367119 779054 391578 539115 551795 034398 307418 402702 986618 039558 610976 689604 911484 367604 196456 812646 662691 590850 327914 363304 630828 821836 528998 234929 878214 712505 510084 488020 452653 868846 426792 081043 430818 282518 029822 713058 419755 900442 463111 524543 256715 017257 849088 074839 082284 580100 056407 965517 228867 633523 092982 360226 977131 424596 436433 942073 164289 857659 773851 459479 178085 101045 272016 823882 680945 639033 505314 274527 281351 426071 896787 755051 254938 557861 760761 466554 735439 443600 > 3¹⁷⁵ [i]

Best Known (172−102, 172, s)-Nets in Base 3

(172−102, 172, 48)-Net over F3 — Constructive and digital

(172−102, 172, 82)-Net over F3 — Digital

(172−102, 172, 296)-Net in Base 3 — Upper bound on s

(172−102, 172, 48)-Net over F₃ — Constructive and digital

(172−102, 172, 82)-Net over F₃ — Digital