Best Known (31−7, 31, s)-Nets in Base 9
(31−7, 31, 19684)-Net over F9 — Constructive and digital
Digital (24, 31, 19684)-net over F9, using
- net defined by OOA [i] based on linear OOA(931, 19684, F9, 7, 7) (dual of [(19684, 7), 137757, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(931, 59053, F9, 7) (dual of [59053, 59022, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(931, 59054, F9, 7) (dual of [59054, 59023, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(931, 59049, F9, 7) (dual of [59049, 59018, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- 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(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(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(931, 59054, F9, 7) (dual of [59054, 59023, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(931, 59053, F9, 7) (dual of [59053, 59022, 8]-code), using
(31−7, 31, 59054)-Net over F9 — Digital
Digital (24, 31, 59054)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(931, 59054, F9, 7) (dual of [59054, 59023, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(931, 59049, F9, 7) (dual of [59049, 59018, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- 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(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(6) ⊂ Ce(5) [i] based on
(31−7, 31, large)-Net in Base 9 — Upper bound on s
There is no (24, 31, large)-net in base 9, because
- 5 times m-reduction [i] would yield (24, 26, large)-net in base 9, but