Best Known (37−10, 37, s)-Nets in Base 27
(37−10, 37, 106289)-Net over F27 — Constructive and digital
Digital (27, 37, 106289)-net over F27, using
- net defined by OOA [i] based on linear OOA(2737, 106289, F27, 10, 10) (dual of [(106289, 10), 1062853, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2737, 531445, F27, 10) (dual of [531445, 531408, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(2737, 531441, F27, 10) (dual of [531441, 531404, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2733, 531441, F27, 9) (dual of [531441, 531408, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(270, 4, F27, 0) (dual of [4, 4, 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(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(2737, 531445, F27, 10) (dual of [531445, 531408, 11]-code), using
(37−10, 37, 399807)-Net over F27 — Digital
Digital (27, 37, 399807)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2737, 399807, F27, 10) (dual of [399807, 399770, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2737, 531441, F27, 10) (dual of [531441, 531404, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(2737, 531441, F27, 10) (dual of [531441, 531404, 11]-code), using
(37−10, 37, large)-Net in Base 27 — Upper bound on s
There is no (27, 37, large)-net in base 27, because
- 8 times m-reduction [i] would yield (27, 29, large)-net in base 27, but