Best Known (64, 64+21, s)-Nets in Base 27
(64, 64+21, 53146)-Net over F27 — Constructive and digital
Digital (64, 85, 53146)-net over F27, using
- 271 times duplication [i] based on digital (63, 84, 53146)-net over F27, using
- net defined by OOA [i] based on linear OOA(2784, 53146, F27, 21, 21) (dual of [(53146, 21), 1115982, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2784, 531461, F27, 21) (dual of [531461, 531377, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(2781, 531442, F27, 21) (dual of [531442, 531361, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2765, 531442, F27, 17) (dual of [531442, 531377, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(273, 19, F27, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,27) or 19-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(2784, 531461, F27, 21) (dual of [531461, 531377, 22]-code), using
- net defined by OOA [i] based on linear OOA(2784, 53146, F27, 21, 21) (dual of [(53146, 21), 1115982, 22]-NRT-code), using
(64, 64+21, 531465)-Net over F27 — Digital
Digital (64, 85, 531465)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2785, 531465, F27, 21) (dual of [531465, 531380, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(2781, 531441, F27, 21) (dual of [531441, 531360, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(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(20) ⊂ Ce(15) [i] based on
(64, 64+21, large)-Net in Base 27 — Upper bound on s
There is no (64, 85, large)-net in base 27, because
- 19 times m-reduction [i] would yield (64, 66, large)-net in base 27, but