Best Known (78−25, 78, s)-Nets in Base 7
(78−25, 78, 204)-Net over F7 — Constructive and digital
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
(78−25, 78, 924)-Net over F7 — Digital
Digital (53, 78, 924)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(778, 924, F7, 25) (dual of [924, 846, 26]-code), using
- 845 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) [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]
- 845 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) [i] based on linear OA(725, 26, F7, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,7)), using
(78−25, 78, 233296)-Net in Base 7 — Upper bound on s
There is no (53, 78, 233297)-net in base 7, because
- 1 times m-reduction [i] would yield (53, 77, 233297)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 118183 113127 802595 427947 904446 814105 129191 815104 451353 294224 741793 > 777 [i]