Best Known (38, 51, s)-Nets in Base 9
(38, 51, 2187)-Net over F9 — Constructive and digital
Digital (38, 51, 2187)-net over F9, using
- 91 times duplication [i] based on digital (37, 50, 2187)-net over F9, using
- net defined by OOA [i] based on linear OOA(950, 2187, F9, 13, 13) (dual of [(2187, 13), 28381, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(950, 13123, F9, 13) (dual of [13123, 13073, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(950, 13124, F9, 13) (dual of [13124, 13074, 14]-code), using
- trace code [i] based on linear OA(8125, 6562, F81, 13) (dual of [6562, 6537, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- trace code [i] based on linear OA(8125, 6562, F81, 13) (dual of [6562, 6537, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(950, 13124, F9, 13) (dual of [13124, 13074, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(950, 13123, F9, 13) (dual of [13123, 13073, 14]-code), using
- net defined by OOA [i] based on linear OOA(950, 2187, F9, 13, 13) (dual of [(2187, 13), 28381, 14]-NRT-code), using
(38, 51, 13128)-Net over F9 — Digital
Digital (38, 51, 13128)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(951, 13128, F9, 13) (dual of [13128, 13077, 14]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(950, 13126, F9, 13) (dual of [13126, 13076, 14]-code), using
- trace code [i] based on linear OA(8125, 6563, F81, 13) (dual of [6563, 6538, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- trace code [i] based on linear OA(8125, 6563, F81, 13) (dual of [6563, 6538, 14]-code), using
- linear OA(950, 13127, F9, 12) (dual of [13127, 13077, 13]-code), using Gilbert–Varšamov bound and bm = 950 > Vbs−1(k−1) = 427086 477803 014263 937998 915140 429097 784258 113009 [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(950, 13126, F9, 13) (dual of [13126, 13076, 14]-code), using
- construction X with Varšamov bound [i] based on
(38, 51, large)-Net in Base 9 — Upper bound on s
There is no (38, 51, large)-net in base 9, because
- 11 times m-reduction [i] would yield (38, 40, large)-net in base 9, but