Best Known (31, 51, s)-Nets in Base 9
(31, 51, 320)-Net over F9 — Constructive and digital
Digital (31, 51, 320)-net over F9, using
- 1 times m-reduction [i] based on digital (31, 52, 320)-net over F9, using
- trace code for nets [i] based on digital (5, 26, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 26, 160)-net over F81, using
(31, 51, 371)-Net over F9 — Digital
Digital (31, 51, 371)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(951, 371, F9, 20) (dual of [371, 320, 21]-code), using
- 94 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 18 times 0, 1, 30 times 0, 1, 37 times 0) [i] based on linear OA(946, 272, F9, 20) (dual of [272, 226, 21]-code), using
- trace code [i] based on linear OA(8123, 136, F81, 20) (dual of [136, 113, 21]-code), using
- extended algebraic-geometric code AGe(F,115P) [i] based on function field F/F81 with g(F) = 3 and N(F) ≥ 136, using
- trace code [i] based on linear OA(8123, 136, F81, 20) (dual of [136, 113, 21]-code), using
- 94 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 18 times 0, 1, 30 times 0, 1, 37 times 0) [i] based on linear OA(946, 272, F9, 20) (dual of [272, 226, 21]-code), using
(31, 51, 41635)-Net in Base 9 — Upper bound on s
There is no (31, 51, 41636)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 4 639023 108725 906466 025276 803726 690144 331748 591553 > 951 [i]