Best Known (64−60, 64, s)-Nets in Base 16
(64−60, 64, 45)-Net over F16 — Constructive and digital
Digital (4, 64, 45)-net over F16, using
- net from sequence [i] based on digital (4, 44)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 4 and N(F) ≥ 45, using
(64−60, 64, 107)-Net over F16 — Upper bound on s (digital)
There is no digital (4, 64, 108)-net over F16, because
- 2 times m-reduction [i] would yield digital (4, 62, 108)-net over F16, but
- extracting embedded orthogonal array [i] would yield linear OA(1662, 108, F16, 58) (dual of [108, 46, 59]-code), but
- construction Y1 [i] would yield
- OA(1661, 65, S16, 58), but
- the linear programming bound shows that M ≥ 159214 122701 309768 707410 104386 945873 298246 228915 255775 554254 178010 880553 254912 / 5487 > 1661 [i]
- linear OA(1646, 108, F16, 43) (dual of [108, 62, 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(1661, 65, S16, 58), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(1662, 108, F16, 58) (dual of [108, 46, 59]-code), but
(64−60, 64, 134)-Net in Base 16 — Upper bound on s
There is no (4, 64, 135)-net in base 16, because
- extracting embedded orthogonal array [i] would yield OA(1664, 135, S16, 60), but
- the linear programming bound shows that M ≥ 927307 381743 870318 843181 626529 192926 580561 440393 777192 457465 581435 826204 073270 718677 661117 803026 211590 373376 / 7 996408 842245 667928 521681 068641 > 1664 [i]