Best Known (54, 54+20, s)-Nets in Base 9
(54, 54+20, 740)-Net over F9 — Constructive and digital
Digital (54, 74, 740)-net over F9, using
- 2 times m-reduction [i] based on digital (54, 76, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 38, 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, 38, 370)-net over F81, using
(54, 54+20, 6581)-Net over F9 — Digital
Digital (54, 74, 6581)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(974, 6581, F9, 20) (dual of [6581, 6507, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(969, 6561, F9, 20) (dual of [6561, 6492, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(953, 6561, F9, 15) (dual of [6561, 6508, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(95, 20, F9, 4) (dual of [20, 15, 5]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
(54, 54+20, 6520494)-Net in Base 9 — Upper bound on s
There is no (54, 74, 6520495)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 41109 891355 904074 809224 259502 316905 392584 536733 821792 588548 829176 895665 > 974 [i]