Best Known (82, 82+37, s)-Nets in Base 9
(82, 82+37, 740)-Net over F9 — Constructive and digital
Digital (82, 119, 740)-net over F9, using
- 13 times m-reduction [i] based on digital (82, 132, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 66, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 66, 370)-net over F81, using
(82, 82+37, 2565)-Net over F9 — Digital
Digital (82, 119, 2565)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9119, 2565, F9, 37) (dual of [2565, 2446, 38]-code), using
- 2445 step Varšamov–Edel lengthening with (ri) = (6, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 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, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 31 times 0, 1, 33 times 0, 1, 35 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 46 times 0, 1, 48 times 0, 1, 51 times 0, 1, 55 times 0, 1, 58 times 0, 1, 63 times 0, 1, 66 times 0, 1, 70 times 0, 1, 75 times 0, 1, 80 times 0, 1, 85 times 0, 1, 91 times 0, 1, 96 times 0, 1, 103 times 0, 1, 109 times 0, 1, 116 times 0, 1, 123 times 0, 1, 132 times 0, 1, 140 times 0, 1, 149 times 0) [i] based on linear OA(937, 38, F9, 37) (dual of [38, 1, 38]-code or 38-arc in PG(36,9)), using
- dual of repetition code with length 38 [i]
- 2445 step Varšamov–Edel lengthening with (ri) = (6, 2, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 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, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 31 times 0, 1, 33 times 0, 1, 35 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 46 times 0, 1, 48 times 0, 1, 51 times 0, 1, 55 times 0, 1, 58 times 0, 1, 63 times 0, 1, 66 times 0, 1, 70 times 0, 1, 75 times 0, 1, 80 times 0, 1, 85 times 0, 1, 91 times 0, 1, 96 times 0, 1, 103 times 0, 1, 109 times 0, 1, 116 times 0, 1, 123 times 0, 1, 132 times 0, 1, 140 times 0, 1, 149 times 0) [i] based on linear OA(937, 38, F9, 37) (dual of [38, 1, 38]-code or 38-arc in PG(36,9)), using
(82, 82+37, 1700697)-Net in Base 9 — Upper bound on s
There is no (82, 119, 1700698)-net in base 9, because
- 1 times m-reduction [i] would yield (82, 118, 1700698)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 39867 387348 313486 917339 061227 299121 362147 729232 067718 258467 135734 024472 355287 629690 239589 850429 386284 891436 407905 > 9118 [i]