Best Known (127−37, 127, s)-Nets in Base 9
(127−37, 127, 768)-Net over F9 — Constructive and digital
Digital (90, 127, 768)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (3, 21, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- digital (69, 106, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 53, 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, 53, 370)-net over F81, using
- digital (3, 21, 28)-net over F9, using
(127−37, 127, 4168)-Net over F9 — Digital
Digital (90, 127, 4168)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9127, 4168, F9, 37) (dual of [4168, 4041, 38]-code), using
- 4040 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, 1, 158 times 0, 1, 168 times 0, 1, 179 times 0, 1, 191 times 0, 1, 202 times 0, 1, 216 times 0, 1, 229 times 0, 1, 244 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]
- 4040 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, 1, 158 times 0, 1, 168 times 0, 1, 179 times 0, 1, 191 times 0, 1, 202 times 0, 1, 216 times 0, 1, 229 times 0, 1, 244 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
(127−37, 127, 4515819)-Net in Base 9 — Upper bound on s
There is no (90, 127, 4515820)-net in base 9, because
- 1 times m-reduction [i] would yield (90, 126, 4515820)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 1 716155 209426 405532 942322 476185 269557 071432 846405 895079 391253 085893 458353 580891 452954 718596 671192 231413 907253 319936 940097 > 9126 [i]