Best Known (66, 66+186, s)-Nets in Base 4
(66, 66+186, 66)-Net over F4 — Constructive and digital
Digital (66, 252, 66)-net over F4, using
- t-expansion [i] based on digital (49, 252, 66)-net over F4, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
- Niederreiter–Xing sequence construction 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]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 49 and N(F) ≥ 66, using
- net from sequence [i] based on digital (49, 65)-sequence over F4, using
(66, 66+186, 99)-Net over F4 — Digital
Digital (66, 252, 99)-net over F4, using
- t-expansion [i] based on digital (61, 252, 99)-net over F4, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 61 and N(F) ≥ 99, using
- net from sequence [i] based on digital (61, 98)-sequence over F4, using
(66, 66+186, 314)-Net over F4 — Upper bound on s (digital)
There is no digital (66, 252, 315)-net over F4, because
- 2 times m-reduction [i] would yield digital (66, 250, 315)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4250, 315, F4, 184) (dual of [315, 65, 185]-code), but
- residual code [i] would yield OA(466, 130, S4, 46), but
- the linear programming bound shows that M ≥ 85 500710 239918 056937 598457 546238 392168 496999 270901 024710 954318 080473 368710 373299 035832 808436 315964 953731 304608 355750 246645 022441 479767 081267 450183 031027 677468 264465 769878 050974 025606 636483 022255 677878 529900 169697 457203 686915 248577 285367 284829 664396 835130 033909 912133 782675 145453 687064 932415 368496 018005 950464 / 14792 016291 163716 498341 420200 257279 735349 377972 186639 808748 146615 449714 019754 184535 021829 772011 820648 802061 890427 812547 915746 041997 531764 615184 271242 411578 612781 720673 055112 499459 082243 654360 383640 447129 126316 017328 820016 161466 487961 088647 669558 103641 211639 601677 558307 > 466 [i]
- residual code [i] would yield OA(466, 130, S4, 46), but
- extracting embedded orthogonal array [i] would yield linear OA(4250, 315, F4, 184) (dual of [315, 65, 185]-code), but
(66, 66+186, 432)-Net in Base 4 — Upper bound on s
There is no (66, 252, 433)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 60 761236 806463 007628 590783 904409 681634 993708 459526 391932 012115 021040 362309 168962 267831 403236 878719 675992 348765 308151 070856 502764 717634 240008 561627 046400 > 4252 [i]