Best Known (37−13, 37, s)-Nets in Base 27
(37−13, 37, 3280)-Net over F27 — Constructive and digital
Digital (24, 37, 3280)-net over F27, using
- net defined by OOA [i] based on linear OOA(2737, 3280, F27, 13, 13) (dual of [(3280, 13), 42603, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2737, 19681, F27, 13) (dual of [19681, 19644, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2737, 19683, F27, 13) (dual of [19683, 19646, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(2737, 19683, F27, 13) (dual of [19683, 19646, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2737, 19681, F27, 13) (dual of [19681, 19644, 14]-code), using
(37−13, 37, 9843)-Net over F27 — Digital
Digital (24, 37, 9843)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2737, 9843, F27, 2, 13) (dual of [(9843, 2), 19649, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2737, 19686, F27, 13) (dual of [19686, 19649, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(2737, 19683, F27, 13) (dual of [19683, 19646, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2734, 19683, F27, 12) (dual of [19683, 19649, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(270, 3, F27, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- OOA 2-folding [i] based on linear OA(2737, 19686, F27, 13) (dual of [19686, 19649, 14]-code), using
(37−13, 37, large)-Net in Base 27 — Upper bound on s
There is no (24, 37, large)-net in base 27, because
- 11 times m-reduction [i] would yield (24, 26, large)-net in base 27, but