Best Known (29, 121, s)-Nets in Base 5
(29, 121, 51)-Net over F5 — Constructive and digital
Digital (29, 121, 51)-net over F5, using
- t-expansion [i] based on digital (22, 121, 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, 121, 56)-Net over F5 — Digital
Digital (29, 121, 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, 121, 237)-Net over F5 — Upper bound on s (digital)
There is no digital (29, 121, 238)-net over F5, because
- extracting embedded orthogonal array [i] would yield linear OA(5121, 238, F5, 92) (dual of [238, 117, 93]-code), but
- construction Y1 [i] 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]
- linear OA(5117, 238, F5, 89) (dual of [238, 121, 90]-code), but
- discarding factors / shortening the dual code would yield linear OA(5117, 226, F5, 89) (dual of [226, 109, 90]-code), but
- construction Y1 [i] would yield
- OA(5116, 143, S5, 89), but
- the linear programming bound shows that M ≥ 2738 237588 593697 454668 762256 522048 887080 570758 264064 445890 203342 535984 543104 632393 806241 452693 939208 984375 / 2 228541 507840 976583 547528 > 5116 [i]
- OA(5109, 226, S5, 83), but
- discarding factors would yield OA(5109, 147, S5, 83), but
- the linear programming bound shows that M ≥ 97832 005097 623965 961965 091455 398251 491670 818930 622769 438002 739592 061328 226246 796901 932611 945085 227489 471435 546875 / 5 803319 642355 292721 815985 390811 721467 > 5109 [i]
- discarding factors would yield OA(5109, 147, S5, 83), but
- OA(5116, 143, S5, 89), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(5117, 226, F5, 89) (dual of [226, 109, 90]-code), but
- OA(5120, 149, S5, 92), but
- construction Y1 [i] would yield
(29, 121, 277)-Net in Base 5 — Upper bound on s
There is no (29, 121, 278)-net in base 5, because
- 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]