Best Known (54, 54+25, s)-Nets in Base 7
(54, 54+25, 204)-Net over F7 — Constructive and digital
Digital (54, 79, 204)-net over F7, using
- 71 times duplication [i] based on digital (53, 78, 204)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (14, 26, 102)-net over F7, using
- trace code for nets [i] based on digital (1, 13, 51)-net over F49, using
- net from sequence [i] based on digital (1, 50)-sequence over F49, using
- trace code for nets [i] based on digital (1, 13, 51)-net over F49, using
- digital (27, 52, 102)-net over F7, using
- trace code for nets [i] based on digital (1, 26, 51)-net over F49, using
- net from sequence [i] based on digital (1, 50)-sequence over F49 (see above)
- trace code for nets [i] based on digital (1, 26, 51)-net over F49, using
- digital (14, 26, 102)-net over F7, using
- (u, u+v)-construction [i] based on
(54, 54+25, 1001)-Net over F7 — Digital
Digital (54, 79, 1001)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(779, 1001, F7, 25) (dual of [1001, 922, 26]-code), using
- 921 step Varšamov–Edel lengthening with (ri) = (5, 2, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 19 times 0, 1, 21 times 0, 1, 23 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 32 times 0, 1, 35 times 0, 1, 39 times 0, 1, 41 times 0, 1, 46 times 0, 1, 49 times 0, 1, 54 times 0, 1, 58 times 0, 1, 64 times 0, 1, 69 times 0, 1, 75 times 0) [i] based on linear OA(725, 26, F7, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,7)), using
- dual of repetition code with length 26 [i]
- 921 step Varšamov–Edel lengthening with (ri) = (5, 2, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 19 times 0, 1, 21 times 0, 1, 23 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 32 times 0, 1, 35 times 0, 1, 39 times 0, 1, 41 times 0, 1, 46 times 0, 1, 49 times 0, 1, 54 times 0, 1, 58 times 0, 1, 64 times 0, 1, 69 times 0, 1, 75 times 0) [i] based on linear OA(725, 26, F7, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,7)), using
(54, 54+25, 274369)-Net in Base 7 — Upper bound on s
There is no (54, 79, 274370)-net in base 7, because
- 1 times m-reduction [i] would yield (54, 78, 274370)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 827294 192297 380817 555337 936776 120145 696025 339035 674740 932885 869209 > 778 [i]