Best Known (82−77, 82, s)-Nets in Base 16
(82−77, 82, 49)-Net over F16 — Constructive and digital
Digital (5, 82, 49)-net over F16, using
- net from sequence [i] based on digital (5, 48)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 5 and N(F) ≥ 49, using
(82−77, 82, 123)-Net over F16 — Upper bound on s (digital)
There is no digital (5, 82, 124)-net over F16, because
- 4 times m-reduction [i] would yield digital (5, 78, 124)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(1678, 124, F16, 73) (dual of [124, 46, 74]-code), but
- construction Y1 [i] would yield
- OA(1677, 81, S16, 73), but
- the linear programming bound shows that M ≥ 20 058253 259194 796242 490750 709498 450326 191022 155255 442417 783937 598455 532112 408827 665059 378347 638784 / 37999 > 1677 [i]
- linear OA(1646, 124, F16, 43) (dual of [124, 78, 44]-code), but
- discarding factors / shortening the dual code would yield linear OA(1646, 97, F16, 43) (dual of [97, 51, 44]-code), but
- construction Y1 [i] would yield
- OA(1645, 49, S16, 43), but
- the linear programming bound shows that M ≥ 79 689768 125026 220634 634045 411816 077548 174434 353547 313152 / 47 > 1645 [i]
- linear OA(1651, 97, F16, 48) (dual of [97, 46, 49]-code), but
- discarding factors / shortening the dual code would yield linear OA(1651, 68, F16, 48) (dual of [68, 17, 49]-code), but
- residual code [i] would yield OA(163, 19, S16, 3), but
- 1 times truncation [i] would yield OA(162, 18, S16, 2), but
- bound for OAs with strength k = 2 [i]
- the Rao or (dual) Hamming bound shows that M ≥ 271 > 162 [i]
- 1 times truncation [i] would yield OA(162, 18, S16, 2), but
- residual code [i] would yield OA(163, 19, S16, 3), but
- discarding factors / shortening the dual code would yield linear OA(1651, 68, F16, 48) (dual of [68, 17, 49]-code), but
- OA(1645, 49, S16, 43), but
- construction Y1 [i] would yield
- discarding factors / shortening the dual code would yield linear OA(1646, 97, F16, 43) (dual of [97, 51, 44]-code), but
- OA(1677, 81, S16, 73), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(1678, 124, F16, 73) (dual of [124, 46, 74]-code), but
(82−77, 82, 124)-Net in Base 16 — Upper bound on s
There is no (5, 82, 125)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(1682, 125, S16, 77), but
- the linear programming bound shows that M ≥ 2161 649133 921743 342607 089169 759846 561746 061907 654428 426051 685627 610431 483177 093803 400073 292688 398087 555122 655311 619583 639552 / 3 798556 074276 596307 580477 > 1682 [i]