Best Known (67, 67+22, s)-Nets in Base 27
(67, 67+22, 48315)-Net over F27 — Constructive and digital
Digital (67, 89, 48315)-net over F27, using
- net defined by OOA [i] based on linear OOA(2789, 48315, F27, 22, 22) (dual of [(48315, 22), 1062841, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(2789, 531465, F27, 22) (dual of [531465, 531376, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(2785, 531441, F27, 22) (dual of [531441, 531356, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(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(21) ⊂ Ce(16) [i] based on
- OA 11-folding and stacking [i] based on linear OA(2789, 531465, F27, 22) (dual of [531465, 531376, 23]-code), using
(67, 67+22, 531465)-Net over F27 — Digital
Digital (67, 89, 531465)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2789, 531465, F27, 22) (dual of [531465, 531376, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(2785, 531441, F27, 22) (dual of [531441, 531356, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(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(21) ⊂ Ce(16) [i] based on
(67, 67+22, large)-Net in Base 27 — Upper bound on s
There is no (67, 89, large)-net in base 27, because
- 20 times m-reduction [i] would yield (67, 69, large)-net in base 27, but