Best Known (52, 52+28, s)-Nets in Base 9
(52, 52+28, 344)-Net over F9 — Constructive and digital
Digital (52, 80, 344)-net over F9, using
- 10 times m-reduction [i] based on digital (52, 90, 344)-net over F9, using
- trace code for nets [i] based on digital (7, 45, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 45, 172)-net over F81, using
(52, 52+28, 932)-Net over F9 — Digital
Digital (52, 80, 932)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(980, 932, F9, 28) (dual of [932, 852, 29]-code), using
- 196 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 7 times 0, 1, 18 times 0, 1, 39 times 0, 1, 57 times 0, 1, 68 times 0) [i] based on linear OA(973, 729, F9, 28) (dual of [729, 656, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 196 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 7 times 0, 1, 18 times 0, 1, 39 times 0, 1, 57 times 0, 1, 68 times 0) [i] based on linear OA(973, 729, F9, 28) (dual of [729, 656, 29]-code), using
(52, 52+28, 214370)-Net in Base 9 — Upper bound on s
There is no (52, 80, 214371)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 21847 911310 037176 063536 279255 285467 276303 902494 831046 511967 736923 709783 687121 > 980 [i]