Best Known (127−44, 127, s)-Nets in Base 9
(127−44, 127, 740)-Net over F9 — Constructive and digital
Digital (83, 127, 740)-net over F9, using
- 7 times m-reduction [i] based on digital (83, 134, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 67, 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, 67, 370)-net over F81, using
(127−44, 127, 1411)-Net over F9 — Digital
Digital (83, 127, 1411)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9127, 1411, F9, 44) (dual of [1411, 1284, 45]-code), using
- 1283 step Varšamov–Edel lengthening with (ri) = (7, 2, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 13 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 36 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 44 times 0, 1, 46 times 0, 1, 49 times 0, 1, 51 times 0, 1, 55 times 0, 1, 57 times 0, 1, 61 times 0, 1, 63 times 0, 1, 68 times 0) [i] based on linear OA(944, 45, F9, 44) (dual of [45, 1, 45]-code or 45-arc in PG(43,9)), using
- dual of repetition code with length 45 [i]
- 1283 step Varšamov–Edel lengthening with (ri) = (7, 2, 2, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 13 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 17 times 0, 1, 18 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 29 times 0, 1, 30 times 0, 1, 32 times 0, 1, 33 times 0, 1, 36 times 0, 1, 37 times 0, 1, 40 times 0, 1, 42 times 0, 1, 44 times 0, 1, 46 times 0, 1, 49 times 0, 1, 51 times 0, 1, 55 times 0, 1, 57 times 0, 1, 61 times 0, 1, 63 times 0, 1, 68 times 0) [i] based on linear OA(944, 45, F9, 44) (dual of [45, 1, 45]-code or 45-arc in PG(43,9)), using
(127−44, 127, 365030)-Net in Base 9 — Upper bound on s
There is no (83, 127, 365031)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 15 445941 383379 721635 961315 786972 610313 661346 609821 897366 556432 936394 879891 171539 256198 561861 275661 064301 502913 672123 268049 > 9127 [i]