Best Known (22, 22+12, s)-Nets in Base 81
(22, 22+12, 88574)-Net over F81 — Constructive and digital
Digital (22, 34, 88574)-net over F81, using
- net defined by OOA [i] based on linear OOA(8134, 88574, F81, 12, 12) (dual of [(88574, 12), 1062854, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8134, 531444, F81, 12) (dual of [531444, 531410, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8131, 531441, F81, 11) (dual of [531441, 531410, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- OA 6-folding and stacking [i] based on linear OA(8134, 531444, F81, 12) (dual of [531444, 531410, 13]-code), using
(22, 22+12, 265722)-Net over F81 — Digital
Digital (22, 34, 265722)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8134, 265722, F81, 2, 12) (dual of [(265722, 2), 531410, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8134, 531444, F81, 12) (dual of [531444, 531410, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8131, 531441, F81, 11) (dual of [531441, 531410, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(8134, 531444, F81, 12) (dual of [531444, 531410, 13]-code), using
(22, 22+12, large)-Net in Base 81 — Upper bound on s
There is no (22, 34, large)-net in base 81, because
- 10 times m-reduction [i] would yield (22, 24, large)-net in base 81, but