Best Known (84, 84+32, s)-Nets in Base 9
(84, 84+32, 770)-Net over F9 — Constructive and digital
Digital (84, 116, 770)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (4, 20, 30)-net over F9, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 4 and N(F) ≥ 30, using
- net from sequence [i] based on digital (4, 29)-sequence over F9, using
- digital (64, 96, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 48, 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, 48, 370)-net over F81, using
- digital (4, 20, 30)-net over F9, using
(84, 84+32, 6576)-Net over F9 — Digital
Digital (84, 116, 6576)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9116, 6576, F9, 32) (dual of [6576, 6460, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- linear OA(9113, 6561, F9, 32) (dual of [6561, 6448, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(9101, 6561, F9, 29) (dual of [6561, 6460, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(93, 15, F9, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
(84, 84+32, 7042167)-Net in Base 9 — Upper bound on s
There is no (84, 116, 7042168)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 492 188742 561882 452931 941580 259366 497536 561716 815145 390299 792525 037195 449196 541880 780151 648214 454250 380833 752065 > 9116 [i]