Best Known (223−161, 223, s)-Nets in Base 4
(223−161, 223, 66)-Net over F4 — Constructive and digital
Digital (62, 223, 66)-net over F4, using
- t-expansion [i] based on digital (49, 223, 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
(223−161, 223, 99)-Net over F4 — Digital
Digital (62, 223, 99)-net over F4, using
- t-expansion [i] based on digital (61, 223, 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
(223−161, 223, 351)-Net over F4 — Upper bound on s (digital)
There is no digital (62, 223, 352)-net over F4, because
- 1 times m-reduction [i] would yield digital (62, 222, 352)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4222, 352, F4, 160) (dual of [352, 130, 161]-code), but
- residual code [i] would yield OA(462, 191, S4, 40), but
- the linear programming bound shows that M ≥ 6 721352 642212 013290 164202 930663 188729 706683 395933 433874 909615 380474 388774 465686 228242 006016 / 300843 081103 152389 559805 379819 997539 174337 778277 134589 > 462 [i]
- residual code [i] would yield OA(462, 191, S4, 40), but
- extracting embedded orthogonal array [i] would yield linear OA(4222, 352, F4, 160) (dual of [352, 130, 161]-code), but
(223−161, 223, 414)-Net in Base 4 — Upper bound on s
There is no (62, 223, 415)-net in base 4, because
- 1 times m-reduction [i] would yield (62, 222, 415)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 45 955264 568487 814613 118464 035157 931206 131110 456132 013771 085334 668230 751518 102925 324196 907498 071813 747809 634025 060553 151485 956148 942110 > 4222 [i]