Best Known (20, 20+6, s)-Nets in Base 9
(20, 20+6, 19684)-Net over F9 — Constructive and digital
Digital (20, 26, 19684)-net over F9, using
- net defined by OOA [i] based on linear OOA(926, 19684, F9, 6, 6) (dual of [(19684, 6), 118078, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(926, 59052, F9, 6) (dual of [59052, 59026, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(926, 59054, F9, 6) (dual of [59054, 59028, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(926, 59049, F9, 6) (dual of [59049, 59023, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(921, 59049, F9, 5) (dual of [59049, 59028, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 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
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(926, 59054, F9, 6) (dual of [59054, 59028, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(926, 59052, F9, 6) (dual of [59052, 59026, 7]-code), using
(20, 20+6, 59054)-Net over F9 — Digital
Digital (20, 26, 59054)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(926, 59054, F9, 6) (dual of [59054, 59028, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(926, 59049, F9, 6) (dual of [59049, 59023, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(921, 59049, F9, 5) (dual of [59049, 59028, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 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
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
(20, 20+6, large)-Net in Base 9 — Upper bound on s
There is no (20, 26, large)-net in base 9, because
- 4 times m-reduction [i] would yield (20, 22, large)-net in base 9, but