Best Known (137, 137+∞, s)-Nets in Base 4
(137, 137+∞, 130)-Net over F4 — Constructive and digital
Digital (137, m, 130)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (137, 129)-sequence over F4, using
- t-expansion [i] based on digital (105, 129)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 105 and N(F) ≥ 130, using
- T7 from the second tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 105 and N(F) ≥ 130, using
- t-expansion [i] based on digital (105, 129)-sequence over F4, using
(137, 137+∞, 176)-Net over F4 — Digital
Digital (137, m, 176)-net over F4 for arbitrarily large m, using
- net from sequence [i] based on digital (137, 175)-sequence over F4, using
- t-expansion [i] based on digital (125, 175)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 125 and N(F) ≥ 176, using
- t-expansion [i] based on digital (125, 175)-sequence over F4, using
(137, 137+∞, 427)-Net in Base 4 — Upper bound on s
There is no (137, m, 428)-net in base 4 for arbitrarily large m, because
- m-reduction [i] would yield (137, 2134, 428)-net in base 4, but
- extracting embedded OOA [i] would yield OOA(42134, 428, S4, 5, 1997), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 250 081453 702432 782820 924513 281021 278511 899600 158641 589314 836080 457847 904386 423263 994010 730280 406090 048213 478218 072994 251751 012841 847716 352900 360066 701121 561507 587248 841509 072005 089903 819941 989908 701071 190050 478741 330509 371003 257050 270968 063340 436111 411239 003151 876119 406007 582326 751759 401566 803709 517180 742275 088162 021089 681787 767910 467110 749610 231752 392933 012968 645379 060537 821196 409365 605056 595105 540967 839375 384930 974381 933287 194419 776597 430051 335060 846094 566750 460454 682468 015826 307098 069934 916417 231793 229719 430307 252480 352257 525783 664537 183804 493926 640803 806778 597120 833245 492267 727442 799533 059091 258601 302487 161590 656435 301315 079710 063412 858284 944022 894866 740147 145789 031793 107326 447327 486598 376369 565994 500682 396539 756164 326169 791091 048554 766504 636014 094342 249115 785627 156808 984810 081943 694104 396979 229900 364804 628244 281826 964298 851774 691188 809903 263049 128264 880276 832117 858069 017506 267762 755746 427219 941165 615602 766123 688312 422236 665148 654563 507349 967884 955580 754375 957121 463382 557220 894698 332464 514451 408355 369307 896762 154498 572323 954563 338294 168616 857038 063971 289687 361844 194673 300011 306311 895687 472128 304413 256092 252605 041433 258199 036536 203977 705032 701809 916725 690965 349932 275230 807622 724139 412120 336890 678193 141439 430500 053127 440596 926340 953098 710385 996186 013336 187688 714240 / 37 > 42134 [i]
- extracting embedded OOA [i] would yield OOA(42134, 428, S4, 5, 1997), but