Best Known (253−184, 253, s)-Nets in Base 4
(253−184, 253, 66)-Net over F4 — Constructive and digital
Digital (69, 253, 66)-net over F4, using
- t-expansion [i] based on digital (49, 253, 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
(253−184, 253, 99)-Net over F4 — Digital
Digital (69, 253, 99)-net over F4, using
- t-expansion [i] based on digital (61, 253, 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
(253−184, 253, 359)-Net over F4 — Upper bound on s (digital)
There is no digital (69, 253, 360)-net over F4, because
- extracting embedded orthogonal array [i] would yield linear OA(4253, 360, F4, 184) (dual of [360, 107, 185]-code), but
- residual code [i] would yield OA(469, 175, S4, 46), but
- the linear programming bound shows that M ≥ 672974 508783 713085 919446 392721 434812 374063 221813 152066 788676 290147 045574 983959 198536 005103 820640 027801 847804 723200 000000 / 1 829712 211075 097010 907956 226624 270704 376891 436886 295247 592787 694007 310519 329949 > 469 [i]
- residual code [i] would yield OA(469, 175, S4, 46), but
(253−184, 253, 456)-Net in Base 4 — Upper bound on s
There is no (69, 253, 457)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 239 475291 039640 876052 704208 738699 309614 630712 611355 664870 711417 456424 291396 762748 019452 654942 155070 003677 767109 620049 875097 880214 169287 413095 068133 358560 > 4253 [i]