Best Known (61, 61+151, s)-Nets in Base 4
(61, 61+151, 66)-Net over F4 — Constructive and digital
Digital (61, 212, 66)-net over F4, using
- t-expansion [i] based on digital (49, 212, 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
(61, 61+151, 99)-Net over F4 — Digital
Digital (61, 212, 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
(61, 61+151, 399)-Net over F4 — Upper bound on s (digital)
There is no digital (61, 212, 400)-net over F4, because
- 3 times m-reduction [i] would yield digital (61, 209, 400)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4209, 400, F4, 148) (dual of [400, 191, 149]-code), but
- residual code [i] would yield OA(461, 251, S4, 37), but
- the linear programming bound shows that M ≥ 107 684832 476558 143648 589731 899624 769127 922671 724172 279174 574834 456723 456000 000000 / 20 050134 070696 232272 809918 735160 283681 235089 > 461 [i]
- residual code [i] would yield OA(461, 251, S4, 37), but
- extracting embedded orthogonal array [i] would yield linear OA(4209, 400, F4, 148) (dual of [400, 191, 149]-code), but
(61, 61+151, 414)-Net in Base 4 — Upper bound on s
There is no (61, 212, 415)-net in base 4, because
- 1 times m-reduction [i] would yield (61, 211, 415)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 12 019447 060020 000718 390251 002994 970478 069011 550280 336929 337808 453126 073834 753997 074809 551626 326622 881292 432097 166700 068914 830848 > 4211 [i]