Best Known (52, 52+28, s)-Nets in Base 27
(52, 52+28, 1406)-Net over F27 — Constructive and digital
Digital (52, 80, 1406)-net over F27, using
- net defined by OOA [i] based on linear OOA(2780, 1406, F27, 28, 28) (dual of [(1406, 28), 39288, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(2780, 19684, F27, 28) (dual of [19684, 19604, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2780, 19687, F27, 28) (dual of [19687, 19607, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2776, 19683, F27, 26) (dual of [19683, 19607, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(271, 4, F27, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 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(2780, 19687, F27, 28) (dual of [19687, 19607, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(2780, 19684, F27, 28) (dual of [19684, 19604, 29]-code), using
(52, 52+28, 9843)-Net over F27 — Digital
Digital (52, 80, 9843)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2780, 9843, F27, 2, 28) (dual of [(9843, 2), 19606, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2780, 19686, F27, 28) (dual of [19686, 19606, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2780, 19687, F27, 28) (dual of [19687, 19607, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(25) [i] based on
- linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2776, 19683, F27, 26) (dual of [19683, 19607, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(271, 4, F27, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 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(2780, 19687, F27, 28) (dual of [19687, 19607, 29]-code), using
- OOA 2-folding [i] based on linear OA(2780, 19686, F27, 28) (dual of [19686, 19606, 29]-code), using
(52, 52+28, large)-Net in Base 27 — Upper bound on s
There is no (52, 80, large)-net in base 27, because
- 26 times m-reduction [i] would yield (52, 54, large)-net in base 27, but