Best Known (72, 72+188, s)-Nets in Base 4
(72, 72+188, 66)-Net over F4 — Constructive and digital
Digital (72, 260, 66)-net over F4, using
- t-expansion [i] based on digital (49, 260, 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
(72, 72+188, 105)-Net over F4 — Digital
Digital (72, 260, 105)-net over F4, using
- t-expansion [i] based on digital (70, 260, 105)-net over F4, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 70 and N(F) ≥ 105, using
- net from sequence [i] based on digital (70, 104)-sequence over F4, using
(72, 72+188, 395)-Net over F4 — Upper bound on s (digital)
There is no digital (72, 260, 396)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4260, 396, F4, 188) (dual of [396, 136, 189]-code), but
- residual code [i] would yield OA(472, 207, S4, 47), but
- 1 times truncation [i] would yield OA(471, 206, S4, 46), but
- the linear programming bound shows that M ≥ 12190 666552 860549 875030 280693 949215 033597 089167 787297 540990 306499 998443 349891 008938 804128 810849 723006 100908 802048 / 2095 122091 727468 263392 248874 603589 540011 988045 238608 933987 894642 253863 > 471 [i]
- 1 times truncation [i] would yield OA(471, 206, S4, 46), but
- residual code [i] would yield OA(472, 207, S4, 47), but
(72, 72+188, 477)-Net in Base 4 — Upper bound on s
There is no (72, 260, 478)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 3 551495 552853 815406 538321 922247 713550 274429 752467 239826 959613 967852 105762 566077 632410 744359 805212 237499 782440 326432 294220 450418 651790 083274 455660 628772 790984 > 4260 [i]