Best Known (40, 61, s)-Nets in Base 81
(40, 61, 53144)-Net over F81 — Constructive and digital
Digital (40, 61, 53144)-net over F81, using
- net defined by OOA [i] based on linear OOA(8161, 53144, F81, 21, 21) (dual of [(53144, 21), 1115963, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- OOA 10-folding and stacking with additional row [i] based on linear OA(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using
(40, 61, 177148)-Net over F81 — Digital
Digital (40, 61, 177148)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8161, 177148, F81, 3, 21) (dual of [(177148, 3), 531383, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8161, 531444, F81, 21) (dual of [531444, 531383, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(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(20) ⊂ Ce(19) [i] based on
- OOA 3-folding [i] based on linear OA(8161, 531444, F81, 21) (dual of [531444, 531383, 22]-code), using
(40, 61, large)-Net in Base 81 — Upper bound on s
There is no (40, 61, large)-net in base 81, because
- 19 times m-reduction [i] would yield (40, 42, large)-net in base 81, but