Best Known (233−169, 233, s)-Nets in Base 4
(233−169, 233, 66)-Net over F4 — Constructive and digital
Digital (64, 233, 66)-net over F4, using
- t-expansion [i] based on digital (49, 233, 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
(233−169, 233, 99)-Net over F4 — Digital
Digital (64, 233, 99)-net over F4, using
- t-expansion [i] based on digital (61, 233, 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
(233−169, 233, 348)-Net over F4 — Upper bound on s (digital)
There is no digital (64, 233, 349)-net over F4, because
- 1 times m-reduction [i] would yield digital (64, 232, 349)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4232, 349, F4, 168) (dual of [349, 117, 169]-code), but
- residual code [i] would yield OA(464, 180, S4, 42), but
- the linear programming bound shows that M ≥ 387317 878288 940952 121032 892106 078533 601878 434569 146042 672353 835001 098548 766344 071344 765591 401529 344000 / 1135 320053 976181 914391 551986 355178 156290 618852 310180 915372 701779 > 464 [i]
- residual code [i] would yield OA(464, 180, S4, 42), but
- extracting embedded orthogonal array [i] would yield linear OA(4232, 349, F4, 168) (dual of [349, 117, 169]-code), but
(233−169, 233, 425)-Net in Base 4 — Upper bound on s
There is no (64, 233, 426)-net in base 4, because
- 1 times m-reduction [i] would yield (64, 232, 426)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 48 267674 940246 676398 589618 071406 220844 214630 233831 149971 421066 573963 068932 128938 860996 060977 524577 226208 183136 340936 648061 232075 011464 640360 > 4232 [i]