Best Known (29, 29+93, s)-Nets in Base 5
(29, 29+93, 51)-Net over F5 — Constructive and digital
Digital (29, 122, 51)-net over F5, using
- t-expansion [i] based on digital (22, 122, 51)-net over F5, using
- net from sequence [i] based on digital (22, 50)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 22 and N(F) ≥ 51, using
- net from sequence [i] based on digital (22, 50)-sequence over F5, using
(29, 29+93, 56)-Net over F5 — Digital
Digital (29, 122, 56)-net over F5, using
- net from sequence [i] based on digital (29, 55)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 29 and N(F) ≥ 56, using
(29, 29+93, 229)-Net over F5 — Upper bound on s (digital)
There is no digital (29, 122, 230)-net over F5, because
- extracting embedded orthogonal array [i] would yield linear OA(5122, 230, F5, 93) (dual of [230, 108, 94]-code), but
- construction Y1 [i] would yield
- OA(5121, 148, S5, 93), but
- the linear programming bound shows that M ≥ 32369 614861 849619 401750 006027 906460 896693 183159 168094 972472 912293 146030 116020 028799 539431 929588 317871 093750 / 5704 050953 652436 468413 > 5121 [i]
- linear OA(5108, 230, F5, 82) (dual of [230, 122, 83]-code), but
- discarding factors / shortening the dual code would yield linear OA(5108, 228, F5, 82) (dual of [228, 120, 83]-code), but
- construction Y1 [i] would yield
- OA(5107, 136, S5, 82), but
- the linear programming bound shows that M ≥ 4 680614 732900 044285 630127 740921 339805 504137 246651 693367 478774 820966 691549 983806 908130 645751 953125 / 7377 156347 408094 956731 > 5107 [i]
- OA(5120, 228, S5, 92), but
- discarding factors would yield OA(5120, 149, S5, 92), but
- the linear programming bound shows that M ≥ 1875 115433 709844 937609 244603 657370 223629 726878 856922 329955 321895 357533 978909 714278 415776 789188 385009 765625 / 2299 458852 249280 370649 > 5120 [i]
- discarding factors would yield OA(5120, 149, S5, 92), but
- OA(5107, 136, S5, 82), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(5108, 228, F5, 82) (dual of [228, 120, 83]-code), but
- OA(5121, 148, S5, 93), but
- construction Y1 [i] would yield
(29, 29+93, 277)-Net in Base 5 — Upper bound on s
There is no (29, 122, 278)-net in base 5, because
- 1 times m-reduction [i] would yield (29, 121, 278)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 4 023653 029074 042490 044441 594173 427807 065601 408794 029906 039425 899070 071537 091422 220225 > 5121 [i]