Best Known (56−28, 56, s)-Nets in Base 81
(56−28, 56, 469)-Net over F81 — Constructive and digital
Digital (28, 56, 469)-net over F81, using
- net defined by OOA [i] based on linear OOA(8156, 469, F81, 28, 28) (dual of [(469, 28), 13076, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(8156, 6566, F81, 28) (dual of [6566, 6510, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(8151, 6561, F81, 26) (dual of [6561, 6510, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 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(27) ⊂ Ce(25) [i] based on
- OA 14-folding and stacking [i] based on linear OA(8156, 6566, F81, 28) (dual of [6566, 6510, 29]-code), using
(56−28, 56, 1997)-Net over F81 — Digital
Digital (28, 56, 1997)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8156, 1997, F81, 3, 28) (dual of [(1997, 3), 5935, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8156, 2188, F81, 3, 28) (dual of [(2188, 3), 6508, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8156, 6564, F81, 28) (dual of [6564, 6508, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8156, 6566, F81, 28) (dual of [6566, 6510, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(8151, 6561, F81, 26) (dual of [6561, 6510, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 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(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(8156, 6566, F81, 28) (dual of [6566, 6510, 29]-code), using
- OOA 3-folding [i] based on linear OA(8156, 6564, F81, 28) (dual of [6564, 6508, 29]-code), using
- discarding factors / shortening the dual code based on linear OOA(8156, 2188, F81, 3, 28) (dual of [(2188, 3), 6508, 29]-NRT-code), using
(56−28, 56, 3253171)-Net in Base 81 — Upper bound on s
There is no (28, 56, 3253172)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 75017 450565 254656 613912 957588 884559 757283 859468 324502 262099 544863 847595 825187 147355 448237 412174 111088 531841 > 8156 [i]