Best Known (19, 19+7, s)-Nets in Base 9
(19, 19+7, 4375)-Net over F9 — Constructive and digital
Digital (19, 26, 4375)-net over F9, using
- net defined by OOA [i] based on linear OOA(926, 4375, F9, 7, 7) (dual of [(4375, 7), 30599, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(926, 13126, F9, 7) (dual of [13126, 13100, 8]-code), using
- trace code [i] based on linear OA(8113, 6563, F81, 7) (dual of [6563, 6550, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(8113, 6561, F81, 7) (dual of [6561, 6548, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(8113, 6563, F81, 7) (dual of [6563, 6550, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(926, 13126, F9, 7) (dual of [13126, 13100, 8]-code), using
(19, 19+7, 13126)-Net over F9 — Digital
Digital (19, 26, 13126)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(926, 13126, F9, 7) (dual of [13126, 13100, 8]-code), using
- trace code [i] based on linear OA(8113, 6563, F81, 7) (dual of [6563, 6550, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(8113, 6561, F81, 7) (dual of [6561, 6548, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(6) ⊂ Ce(5) [i] based on
- trace code [i] based on linear OA(8113, 6563, F81, 7) (dual of [6563, 6550, 8]-code), using
(19, 19+7, large)-Net in Base 9 — Upper bound on s
There is no (19, 26, large)-net in base 9, because
- 5 times m-reduction [i] would yield (19, 21, large)-net in base 9, but