Best Known (69, 69+25, s)-Nets in Base 7
(69, 69+25, 688)-Net over F7 — Constructive and digital
Digital (69, 94, 688)-net over F7, using
- 2 times m-reduction [i] based on digital (69, 96, 688)-net over F7, using
- trace code for nets [i] based on digital (21, 48, 344)-net over F49, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- the Hermitian function field over F49 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 21 and N(F) ≥ 344, using
- net from sequence [i] based on digital (21, 343)-sequence over F49, using
- trace code for nets [i] based on digital (21, 48, 344)-net over F49, using
(69, 69+25, 3348)-Net over F7 — Digital
Digital (69, 94, 3348)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(794, 3348, F7, 25) (dual of [3348, 3254, 26]-code), using
- 3253 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, 1, 81 times 0, 1, 89 times 0, 1, 96 times 0, 1, 104 times 0, 1, 114 times 0, 1, 123 times 0, 1, 134 times 0, 1, 145 times 0, 1, 158 times 0, 1, 171 times 0, 1, 186 times 0, 1, 201 times 0, 1, 219 times 0, 1, 238 times 0, 1, 258 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]
- 3253 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, 1, 81 times 0, 1, 89 times 0, 1, 96 times 0, 1, 104 times 0, 1, 114 times 0, 1, 123 times 0, 1, 134 times 0, 1, 145 times 0, 1, 158 times 0, 1, 171 times 0, 1, 186 times 0, 1, 201 times 0, 1, 219 times 0, 1, 238 times 0, 1, 258 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
(69, 69+25, 3124056)-Net in Base 7 — Upper bound on s
There is no (69, 94, 3124057)-net in base 7, because
- 1 times m-reduction [i] would yield (69, 93, 3124057)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 3 927515 813695 612133 706041 066277 823645 658686 315888 249590 468074 832357 643267 877089 > 793 [i]