Best Known (231−167, 231, s)-Nets in Base 4
(231−167, 231, 66)-Net over F4 — Constructive and digital
Digital (64, 231, 66)-net over F4, using
- t-expansion [i] based on digital (49, 231, 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
(231−167, 231, 99)-Net over F4 — Digital
Digital (64, 231, 99)-net over F4, using
- t-expansion [i] based on digital (61, 231, 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
(231−167, 231, 367)-Net over F4 — Upper bound on s (digital)
There is no digital (64, 231, 368)-net over F4, because
- 3 times m-reduction [i] would yield digital (64, 228, 368)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4228, 368, F4, 164) (dual of [368, 140, 165]-code), but
- residual code [i] would yield OA(464, 203, S4, 41), but
- the linear programming bound shows that M ≥ 9237 828678 014661 888733 201438 074087 629177 850315 553208 766694 988543 356653 154675 278877 490717 655040 / 25 848321 162508 053807 967718 560003 483887 415061 992757 599441 > 464 [i]
- residual code [i] would yield OA(464, 203, S4, 41), but
- extracting embedded orthogonal array [i] would yield linear OA(4228, 368, F4, 164) (dual of [368, 140, 165]-code), but
(231−167, 231, 427)-Net in Base 4 — Upper bound on s
There is no (64, 231, 428)-net in base 4, because
- 1 times m-reduction [i] would yield (64, 230, 428)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 3 393703 379457 899000 488810 695152 194741 895706 416790 709474 352050 217833 299465 405474 259306 438970 872609 280393 798183 507127 230677 400556 756378 430300 > 4230 [i]