Best Known (69−17, 69, s)-Nets in Base 27
(69−17, 69, 66433)-Net over F27 — Constructive and digital
Digital (52, 69, 66433)-net over F27, using
- net defined by OOA [i] based on linear OOA(2769, 66433, F27, 17, 17) (dual of [(66433, 17), 1129292, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2769, 531465, F27, 17) (dual of [531465, 531396, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(2765, 531441, F27, 17) (dual of [531441, 531376, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2745, 531441, F27, 12) (dual of [531441, 531396, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(16) ⊂ Ce(11) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(2769, 531465, F27, 17) (dual of [531465, 531396, 18]-code), using
(69−17, 69, 531465)-Net over F27 — Digital
Digital (52, 69, 531465)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2769, 531465, F27, 17) (dual of [531465, 531396, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(2765, 531441, F27, 17) (dual of [531441, 531376, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(2745, 531441, F27, 12) (dual of [531441, 531396, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(16) ⊂ Ce(11) [i] based on
(69−17, 69, large)-Net in Base 27 — Upper bound on s
There is no (52, 69, large)-net in base 27, because
- 15 times m-reduction [i] would yield (52, 54, large)-net in base 27, but