Best Known (31−11, 31, s)-Nets in Base 27
(31−11, 31, 3937)-Net over F27 — Constructive and digital
Digital (20, 31, 3937)-net over F27, using
- net defined by OOA [i] based on linear OOA(2731, 3937, F27, 11, 11) (dual of [(3937, 11), 43276, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2731, 19686, F27, 11) (dual of [19686, 19655, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(2731, 19683, F27, 11) (dual of [19683, 19652, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2728, 19683, F27, 10) (dual of [19683, 19655, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(2731, 19686, F27, 11) (dual of [19686, 19655, 12]-code), using
(31−11, 31, 9843)-Net over F27 — Digital
Digital (20, 31, 9843)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2731, 9843, F27, 2, 11) (dual of [(9843, 2), 19655, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2731, 19686, F27, 11) (dual of [19686, 19655, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(2731, 19683, F27, 11) (dual of [19683, 19652, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2728, 19683, F27, 10) (dual of [19683, 19655, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- OOA 2-folding [i] based on linear OA(2731, 19686, F27, 11) (dual of [19686, 19655, 12]-code), using
(31−11, 31, large)-Net in Base 27 — Upper bound on s
There is no (20, 31, large)-net in base 27, because
- 9 times m-reduction [i] would yield (20, 22, large)-net in base 27, but