Best Known (59, 59+166, s)-Nets in Base 4
(59, 59+166, 66)-Net over F4 — Constructive and digital
Digital (59, 225, 66)-net over F4, using
- t-expansion [i] based on digital (49, 225, 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
(59, 59+166, 91)-Net over F4 — Digital
Digital (59, 225, 91)-net over F4, using
- t-expansion [i] based on digital (50, 225, 91)-net over F4, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 50 and N(F) ≥ 91, using
- net from sequence [i] based on digital (50, 90)-sequence over F4, using
(59, 59+166, 289)-Net over F4 — Upper bound on s (digital)
There is no digital (59, 225, 290)-net over F4, because
- 2 times m-reduction [i] would yield digital (59, 223, 290)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4223, 290, F4, 164) (dual of [290, 67, 165]-code), but
- residual code [i] would yield OA(459, 125, S4, 41), but
- the linear programming bound shows that M ≥ 6 098659 704705 486904 667441 402011 989050 548288 910427 912825 765842 283670 760248 415348 522730 039189 577335 282788 475369 967101 883338 375933 306396 897634 126855 141698 525692 111751 859853 590528 / 17 865783 539531 952003 937094 580988 623442 037272 256331 005047 247373 356459 142621 652981 139301 072409 181322 127802 625025 609129 027701 857126 610015 951793 > 459 [i]
- residual code [i] would yield OA(459, 125, S4, 41), but
- extracting embedded orthogonal array [i] would yield linear OA(4223, 290, F4, 164) (dual of [290, 67, 165]-code), but
(59, 59+166, 388)-Net in Base 4 — Upper bound on s
There is no (59, 225, 389)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 3466 944274 565295 706479 264273 478653 459859 761202 086219 025037 262633 098500 687956 445451 505166 236769 386281 595841 365090 901259 069304 274464 220000 > 4225 [i]