Best Known (50, 50+26, s)-Nets in Base 81
(50, 50+26, 40880)-Net over F81 — Constructive and digital
Digital (50, 76, 40880)-net over F81, using
- net defined by OOA [i] based on linear OOA(8176, 40880, F81, 26, 26) (dual of [(40880, 26), 1062804, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(8176, 531440, F81, 26) (dual of [531440, 531364, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8176, 531441, F81, 26) (dual of [531441, 531365, 27]-code), using
- an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- discarding factors / shortening the dual code based on linear OA(8176, 531441, F81, 26) (dual of [531441, 531365, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(8176, 531440, F81, 26) (dual of [531440, 531364, 27]-code), using
(50, 50+26, 177148)-Net over F81 — Digital
Digital (50, 76, 177148)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8176, 177148, F81, 3, 26) (dual of [(177148, 3), 531368, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8176, 531444, F81, 26) (dual of [531444, 531368, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(8176, 531441, F81, 26) (dual of [531441, 531365, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8173, 531441, F81, 25) (dual of [531441, 531368, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(25) ⊂ Ce(24) [i] based on
- OOA 3-folding [i] based on linear OA(8176, 531444, F81, 26) (dual of [531444, 531368, 27]-code), using
(50, 50+26, large)-Net in Base 81 — Upper bound on s
There is no (50, 76, large)-net in base 81, because
- 24 times m-reduction [i] would yield (50, 52, large)-net in base 81, but