Best Known (66−16, 66, s)-Nets in Base 27
(66−16, 66, 66433)-Net over F27 — Constructive and digital
Digital (50, 66, 66433)-net over F27, using
- 271 times duplication [i] based on digital (49, 65, 66433)-net over F27, using
- net defined by OOA [i] based on linear OOA(2765, 66433, F27, 16, 16) (dual of [(66433, 16), 1062863, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2765, 531464, F27, 16) (dual of [531464, 531399, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2765, 531465, F27, 16) (dual of [531465, 531400, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(2761, 531441, F27, 16) (dual of [531441, 531380, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(2741, 531441, F27, 11) (dual of [531441, 531400, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(274, 24, F27, 4) (dual of [24, 20, 5]-code or 24-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(2765, 531465, F27, 16) (dual of [531465, 531400, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2765, 531464, F27, 16) (dual of [531464, 531399, 17]-code), using
- net defined by OOA [i] based on linear OOA(2765, 66433, F27, 16, 16) (dual of [(66433, 16), 1062863, 17]-NRT-code), using
(66−16, 66, 531469)-Net over F27 — Digital
Digital (50, 66, 531469)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2766, 531469, F27, 16) (dual of [531469, 531403, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- linear OA(2761, 531441, F27, 16) (dual of [531441, 531380, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 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(275, 28, F27, 5) (dual of [28, 23, 6]-code or 28-arc in PG(4,27)), using
- extended Reed–Solomon code RSe(23,27) [i]
- the expurgated narrow-sense BCH-code C(I) with length 28 | 272−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
(66−16, 66, large)-Net in Base 27 — Upper bound on s
There is no (50, 66, large)-net in base 27, because
- 14 times m-reduction [i] would yield (50, 52, large)-net in base 27, but