Best Known (72, 72+179, s)-Nets in Base 4
(72, 72+179, 66)-Net over F4 — Constructive and digital
Digital (72, 251, 66)-net over F4, using
- t-expansion [i] based on digital (49, 251, 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+179, 105)-Net over F4 — Digital
Digital (72, 251, 105)-net over F4, using
- t-expansion [i] based on digital (70, 251, 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+179, 446)-Net over F4 — Upper bound on s (digital)
There is no digital (72, 251, 447)-net over F4, because
- 3 times m-reduction [i] would yield digital (72, 248, 447)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4248, 447, F4, 176) (dual of [447, 199, 177]-code), but
- residual code [i] would yield OA(472, 270, S4, 44), but
- the linear programming bound shows that M ≥ 786435 471178 717771 595724 051341 292858 017461 789380 078325 091944 259711 116987 003218 742476 800000 000000 / 34141 957272 513139 588342 951169 793634 345796 839121 212019 > 472 [i]
- residual code [i] would yield OA(472, 270, S4, 44), but
- extracting embedded orthogonal array [i] would yield linear OA(4248, 447, F4, 176) (dual of [447, 199, 177]-code), but
(72, 72+179, 485)-Net in Base 4 — Upper bound on s
There is no (72, 251, 486)-net in base 4, because
- 1 times m-reduction [i] would yield (72, 250, 486)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 3 720932 680378 735408 075822 229969 604520 050154 446276 113660 021746 678807 226582 687835 746837 789649 821629 007819 420908 643216 663342 809589 142126 220696 502036 362108 > 4250 [i]