Best Known (35, 53, s)-Nets in Base 27
(35, 53, 2187)-Net over F27 — Constructive and digital
Digital (35, 53, 2187)-net over F27, using
- 271 times duplication [i] based on digital (34, 52, 2187)-net over F27, using
- net defined by OOA [i] based on linear OOA(2752, 2187, F27, 18, 18) (dual of [(2187, 18), 39314, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2752, 19683, F27, 18) (dual of [19683, 19631, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- OA 9-folding and stacking [i] based on linear OA(2752, 19683, F27, 18) (dual of [19683, 19631, 19]-code), using
- net defined by OOA [i] based on linear OOA(2752, 2187, F27, 18, 18) (dual of [(2187, 18), 39314, 19]-NRT-code), using
(35, 53, 11728)-Net over F27 — Digital
Digital (35, 53, 11728)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2753, 11728, F27, 18) (dual of [11728, 11675, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2753, 19690, F27, 18) (dual of [19690, 19637, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(2752, 19683, F27, 18) (dual of [19683, 19631, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(2746, 19683, F27, 16) (dual of [19683, 19637, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 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(17) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2753, 19690, F27, 18) (dual of [19690, 19637, 19]-code), using
(35, 53, large)-Net in Base 27 — Upper bound on s
There is no (35, 53, large)-net in base 27, because
- 16 times m-reduction [i] would yield (35, 37, large)-net in base 27, but