Best Known (25−9, 25, s)-Nets in Base 81
(25−9, 25, 132860)-Net over F81 — Constructive and digital
Digital (16, 25, 132860)-net over F81, using
- net defined by OOA [i] based on linear OOA(8125, 132860, F81, 9, 9) (dual of [(132860, 9), 1195715, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(8125, 531441, F81, 9) (dual of [531441, 531416, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(8125, 531441, F81, 9) (dual of [531441, 531416, 10]-code), using
(25−9, 25, 265722)-Net over F81 — Digital
Digital (16, 25, 265722)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8125, 265722, F81, 2, 9) (dual of [(265722, 2), 531419, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8125, 531444, F81, 9) (dual of [531444, 531419, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(8125, 531441, F81, 9) (dual of [531441, 531416, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- OOA 2-folding [i] based on linear OA(8125, 531444, F81, 9) (dual of [531444, 531419, 10]-code), using
(25−9, 25, large)-Net in Base 81 — Upper bound on s
There is no (16, 25, large)-net in base 81, because
- 7 times m-reduction [i] would yield (16, 18, large)-net in base 81, but