Best Known (64, 64+24, s)-Nets in Base 9
(64, 64+24, 740)-Net over F9 — Constructive and digital
Digital (64, 88, 740)-net over F9, using
- 8 times m-reduction [i] based on 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
(64, 64+24, 6576)-Net over F9 — Digital
Digital (64, 88, 6576)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(988, 6576, F9, 24) (dual of [6576, 6488, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(985, 6561, F9, 24) (dual of [6561, 6476, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(973, 6561, F9, 21) (dual of [6561, 6488, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(23) ⊂ Ce(20) [i] based on
(64, 64+24, 6577325)-Net in Base 9 — Upper bound on s
There is no (64, 88, 6577326)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 940461 108397 738608 900648 936910 484040 341960 018788 945971 277579 223095 609926 150399 348161 > 988 [i]