Best Known (59−20, 59, s)-Nets in Base 81
(59−20, 59, 53144)-Net over F81 — Constructive and digital
Digital (39, 59, 53144)-net over F81, using
- 811 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(8158, 53144, F81, 20, 20) (dual of [(53144, 20), 1062822, 21]-NRT-code), using
(59−20, 59, 224842)-Net over F81 — Digital
Digital (39, 59, 224842)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8159, 224842, F81, 2, 20) (dual of [(224842, 2), 449625, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8159, 265724, F81, 2, 20) (dual of [(265724, 2), 531389, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8159, 531448, F81, 20) (dual of [531448, 531389, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [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(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(8159, 531448, F81, 20) (dual of [531448, 531389, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(8159, 265724, F81, 2, 20) (dual of [(265724, 2), 531389, 21]-NRT-code), using
(59−20, 59, large)-Net in Base 81 — Upper bound on s
There is no (39, 59, large)-net in base 81, because
- 18 times m-reduction [i] would yield (39, 41, large)-net in base 81, but