Best Known (54−21, 54, s)-Nets in Base 9
(54−21, 54, 320)-Net over F9 — Constructive and digital
Digital (33, 54, 320)-net over F9, using
- 2 times m-reduction [i] based on digital (33, 56, 320)-net over F9, using
- trace code for nets [i] based on digital (5, 28, 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, 28, 160)-net over F81, using
(54−21, 54, 402)-Net over F9 — Digital
Digital (33, 54, 402)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(954, 402, F9, 21) (dual of [402, 348, 22]-code), using
- 66 step Varšamov–Edel lengthening with (ri) = (1, 26 times 0, 1, 38 times 0) [i] based on linear OA(952, 334, F9, 21) (dual of [334, 282, 22]-code), using
- trace code [i] based on linear OA(8126, 167, F81, 21) (dual of [167, 141, 22]-code), using
- extended algebraic-geometric code AGe(F,145P) [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 167, using
- trace code [i] based on linear OA(8126, 167, F81, 21) (dual of [167, 141, 22]-code), using
- 66 step Varšamov–Edel lengthening with (ri) = (1, 26 times 0, 1, 38 times 0) [i] based on linear OA(952, 334, F9, 21) (dual of [334, 282, 22]-code), using
(54−21, 54, 64614)-Net in Base 9 — Upper bound on s
There is no (33, 54, 64615)-net in base 9, because
- 1 times m-reduction [i] would yield (33, 53, 64615)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 375 713472 335166 624613 699442 340625 388574 283235 573553 > 953 [i]