Best Known (7, 7+97, s)-Nets in Base 16
(7, 7+97, 65)-Net over F16 — Constructive and digital
Digital (7, 104, 65)-net over F16, using
- t-expansion [i] based on digital (6, 104, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
(7, 7+97, 161)-Net over F16 — Upper bound on s (digital)
There is no digital (7, 104, 162)-net over F16, because
- 1 times m-reduction [i] would yield digital (7, 103, 162)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(16103, 162, F16, 96) (dual of [162, 59, 97]-code), but
- construction Y1 [i] would yield
- linear OA(16102, 107, F16, 96) (dual of [107, 5, 97]-code), but
- construction Y1 [i] would yield
- OA(16101, 103, S16, 96), but
- the (dual) Plotkin bound shows that M ≥ 4627 391781 531740 192663 407156 229397 278798 832780 749968 534992 541566 920840 538654 179420 776324 473078 006993 933138 484175 163508 129792 / 97 > 16101 [i]
- OA(165, 107, S16, 4), but
- discarding factors would yield OA(165, 97, S16, 4), but
- the Rao or (dual) Hamming bound shows that M ≥ 1 049056 > 165 [i]
- discarding factors would yield OA(165, 97, S16, 4), but
- OA(16101, 103, S16, 96), but
- construction Y1 [i] would yield
- OA(1659, 162, S16, 55), but
- discarding factors would yield OA(1659, 158, S16, 55), but
- the linear programming bound shows that M ≥ 1 751872 128079 872472 547941 838539 530895 802398 111785 402312 961189 355293 686055 441948 303058 733784 687094 938824 867840 / 15 835355 243697 262421 313219 417467 218513 > 1659 [i]
- discarding factors would yield OA(1659, 158, S16, 55), but
- linear OA(16102, 107, F16, 96) (dual of [107, 5, 97]-code), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(16103, 162, F16, 96) (dual of [162, 59, 97]-code), but
(7, 7+97, 165)-Net in Base 16 — Upper bound on s
There is no (7, 104, 166)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(16104, 166, S16, 97), but
- the linear programming bound shows that M ≥ 198774 673472 859573 888449 875624 130725 596937 069650 806338 386835 560086 995299 036746 720773 426027 709058 051566 059766 921210 262701 746636 822201 696317 837646 099676 931723 197706 876319 956992 / 1 145683 417871 470513 572885 574541 180812 228234 357097 > 16104 [i]