Best Known (46, 46+24, s)-Nets in Base 81
(46, 46+24, 44287)-Net over F81 — Constructive and digital
Digital (46, 70, 44287)-net over F81, using
- net defined by OOA [i] based on linear OOA(8170, 44287, F81, 24, 24) (dual of [(44287, 24), 1062818, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(8170, 531444, F81, 24) (dual of [531444, 531374, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(8170, 531441, F81, 24) (dual of [531441, 531371, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(8167, 531441, F81, 23) (dual of [531441, 531374, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 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(23) ⊂ Ce(22) [i] based on
- OA 12-folding and stacking [i] based on linear OA(8170, 531444, F81, 24) (dual of [531444, 531374, 25]-code), using
(46, 46+24, 177148)-Net over F81 — Digital
Digital (46, 70, 177148)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8170, 177148, F81, 3, 24) (dual of [(177148, 3), 531374, 25]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8170, 531444, F81, 24) (dual of [531444, 531374, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(8170, 531441, F81, 24) (dual of [531441, 531371, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(8167, 531441, F81, 23) (dual of [531441, 531374, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 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(23) ⊂ Ce(22) [i] based on
- OOA 3-folding [i] based on linear OA(8170, 531444, F81, 24) (dual of [531444, 531374, 25]-code), using
(46, 46+24, large)-Net in Base 81 — Upper bound on s
There is no (46, 70, large)-net in base 81, because
- 22 times m-reduction [i] would yield (46, 48, large)-net in base 81, but