Best Known (51, 51+26, s)-Nets in Base 81
(51, 51+26, 40880)-Net over F81 — Constructive and digital
Digital (51, 77, 40880)-net over F81, using
- 811 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(8176, 40880, F81, 26, 26) (dual of [(40880, 26), 1062804, 27]-NRT-code), using
(51, 51+26, 197089)-Net over F81 — Digital
Digital (51, 77, 197089)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8177, 197089, F81, 2, 26) (dual of [(197089, 2), 394101, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8177, 265724, F81, 2, 26) (dual of [(265724, 2), 531371, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8177, 531448, F81, 26) (dual of [531448, 531371, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(23) [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(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(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(25) ⊂ Ce(23) [i] based on
- OOA 2-folding [i] based on linear OA(8177, 531448, F81, 26) (dual of [531448, 531371, 27]-code), using
- discarding factors / shortening the dual code based on linear OOA(8177, 265724, F81, 2, 26) (dual of [(265724, 2), 531371, 27]-NRT-code), using
(51, 51+26, large)-Net in Base 81 — Upper bound on s
There is no (51, 77, large)-net in base 81, because
- 24 times m-reduction [i] would yield (51, 53, large)-net in base 81, but