Best Known (73−25, 73, s)-Nets in Base 81
(73−25, 73, 44286)-Net over F81 — Constructive and digital
Digital (48, 73, 44286)-net over F81, using
- net defined by OOA [i] based on linear OOA(8173, 44286, F81, 25, 25) (dual of [(44286, 25), 1107077, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8173, 531433, F81, 25) (dual of [531433, 531360, 26]-code), using
- discarding factors / shortening the dual code based on 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]
- discarding factors / shortening the dual code based on linear OA(8173, 531441, F81, 25) (dual of [531441, 531368, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8173, 531433, F81, 25) (dual of [531433, 531360, 26]-code), using
(73−25, 73, 177148)-Net over F81 — Digital
Digital (48, 73, 177148)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8173, 177148, F81, 3, 25) (dual of [(177148, 3), 531371, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8173, 531444, F81, 25) (dual of [531444, 531371, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- 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(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(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(24) ⊂ Ce(23) [i] based on
- OOA 3-folding [i] based on linear OA(8173, 531444, F81, 25) (dual of [531444, 531371, 26]-code), using
(73−25, 73, large)-Net in Base 81 — Upper bound on s
There is no (48, 73, large)-net in base 81, because
- 23 times m-reduction [i] would yield (48, 50, large)-net in base 81, but