Best Known (79, 79+160, s)-Nets in Base 3
(79, 79+160, 54)-Net over F3 — Constructive and digital
Digital (79, 239, 54)-net over F3, using
- net from sequence [i] based on digital (79, 53)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 53)-sequence over F9, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 13 and N(F) ≥ 64, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- base reduction for sequences [i] based on digital (13, 53)-sequence over F9, using
(79, 79+160, 84)-Net over F3 — Digital
Digital (79, 239, 84)-net over F3, using
- t-expansion [i] based on digital (71, 239, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(79, 79+160, 249)-Net over F3 — Upper bound on s (digital)
There is no digital (79, 239, 250)-net over F3, because
- 1 times m-reduction [i] would yield digital (79, 238, 250)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3238, 250, F3, 159) (dual of [250, 12, 160]-code), but
- residual code [i] would yield OA(379, 90, S3, 53), but
- 1 times truncation [i] would yield OA(378, 89, S3, 52), but
- the linear programming bound shows that M ≥ 298 630446 198644 459587 118144 486403 305957 001183 / 16 135903 > 378 [i]
- 1 times truncation [i] would yield OA(378, 89, S3, 52), but
- residual code [i] would yield OA(379, 90, S3, 53), but
- extracting embedded orthogonal array [i] would yield linear OA(3238, 250, F3, 159) (dual of [250, 12, 160]-code), but
(79, 79+160, 333)-Net in Base 3 — Upper bound on s
There is no (79, 239, 334)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1 125006 259326 215529 986329 071855 188027 615217 983098 475999 638574 618908 937390 275451 705031 941962 332951 628183 630601 037345 > 3239 [i]