Best Known (38, 38+20, s)-Nets in Base 81
(38, 38+20, 53144)-Net over F81 — Constructive and digital
Digital (38, 58, 53144)-net over F81, using
- net defined by OOA [i] based on linear OOA(8158, 53144, F81, 20, 20) (dual of [(53144, 20), 1062822, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8158, 531440, F81, 20) (dual of [531440, 531382, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(8158, 531440, F81, 20) (dual of [531440, 531382, 21]-code), using
(38, 38+20, 177148)-Net over F81 — Digital
Digital (38, 58, 177148)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8158, 177148, F81, 3, 20) (dual of [(177148, 3), 531386, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8158, 531444, F81, 20) (dual of [531444, 531386, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8155, 531441, F81, 19) (dual of [531441, 531386, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- OOA 3-folding [i] based on linear OA(8158, 531444, F81, 20) (dual of [531444, 531386, 21]-code), using
(38, 38+20, large)-Net in Base 81 — Upper bound on s
There is no (38, 58, large)-net in base 81, because
- 18 times m-reduction [i] would yield (38, 40, large)-net in base 81, but