Best Known (63, 86, s)-Nets in Base 9
(63, 86, 740)-Net over F9 — Constructive and digital
Digital (63, 86, 740)-net over F9, using
- 8 times m-reduction [i] based on digital (63, 94, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 47, 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, 47, 370)-net over F81, using
(63, 86, 6584)-Net over F9 — Digital
Digital (63, 86, 6584)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(986, 6584, F9, 23) (dual of [6584, 6498, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(985, 6582, F9, 23) (dual of [6582, 6497, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- linear OA(981, 6562, F9, 23) (dual of [6562, 6481, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(965, 6562, F9, 19) (dual of [6562, 6497, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(94, 20, F9, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,9)), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- linear OA(985, 6583, F9, 22) (dual of [6583, 6498, 23]-code), using Gilbert–Varšamov bound and bm = 985 > Vbs−1(k−1) = 26 814169 877549 515474 861916 378537 977998 472229 181722 664257 705239 781331 415610 473329 [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(985, 6582, F9, 23) (dual of [6582, 6497, 24]-code), using
- construction X with Varšamov bound [i] based on
(63, 86, large)-Net in Base 9 — Upper bound on s
There is no (63, 86, large)-net in base 9, because
- 21 times m-reduction [i] would yield (63, 65, large)-net in base 9, but