Best Known (59, 59+22, s)-Nets in Base 9
(59, 59+22, 740)-Net over F9 — Constructive and digital
Digital (59, 81, 740)-net over F9, using
- 5 times m-reduction [i] based on digital (59, 86, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 43, 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, 43, 370)-net over F81, using
(59, 59+22, 6578)-Net over F9 — Digital
Digital (59, 81, 6578)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(981, 6578, F9, 22) (dual of [6578, 6497, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(980, 6576, F9, 22) (dual of [6576, 6496, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(977, 6561, F9, 22) (dual of [6561, 6484, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(965, 6561, F9, 19) (dual of [6561, 6496, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- linear OA(980, 6577, F9, 21) (dual of [6577, 6497, 22]-code), using Gilbert–Varšamov bound and bm = 980 > Vbs−1(k−1) = 10532 133580 878318 518312 656330 986688 734757 347975 912796 678268 845108 821558 040449 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(980, 6576, F9, 22) (dual of [6576, 6496, 23]-code), using
- construction X with Varšamov bound [i] based on
(59, 59+22, 6525572)-Net in Base 9 — Upper bound on s
There is no (59, 81, 6525573)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 196627 375313 128479 649453 901502 586193 610180 293244 092093 787096 098108 809273 252345 > 981 [i]