Best Known (14, 14+8, s)-Nets in Base 81
(14, 14+8, 132861)-Net over F81 — Constructive and digital
Digital (14, 22, 132861)-net over F81, using
- net defined by OOA [i] based on linear OOA(8122, 132861, F81, 8, 8) (dual of [(132861, 8), 1062866, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8122, 531444, F81, 8) (dual of [531444, 531422, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(8119, 531441, F81, 7) (dual of [531441, 531422, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- OA 4-folding and stacking [i] based on linear OA(8122, 531444, F81, 8) (dual of [531444, 531422, 9]-code), using
(14, 14+8, 265722)-Net over F81 — Digital
Digital (14, 22, 265722)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8122, 265722, F81, 2, 8) (dual of [(265722, 2), 531422, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8122, 531444, F81, 8) (dual of [531444, 531422, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 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(8119, 531441, F81, 7) (dual of [531441, 531422, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(7) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(8122, 531444, F81, 8) (dual of [531444, 531422, 9]-code), using
(14, 14+8, large)-Net in Base 81 — Upper bound on s
There is no (14, 22, large)-net in base 81, because
- 6 times m-reduction [i] would yield (14, 16, large)-net in base 81, but