Best Known (53, 53+17, s)-Nets in Base 27
(53, 53+17, 66433)-Net over F27 — Constructive and digital
Digital (53, 70, 66433)-net over F27, using
- 271 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(2769, 66433, F27, 17, 17) (dual of [(66433, 17), 1129292, 18]-NRT-code), using
(53, 53+17, 531470)-Net over F27 — Digital
Digital (53, 70, 531470)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2770, 531470, F27, 17) (dual of [531470, 531400, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- 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(2741, 531442, F27, 11) (dual of [531442, 531401, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [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 C([0,8]) ⊂ C([0,5]) [i] based on
(53, 53+17, large)-Net in Base 27 — Upper bound on s
There is no (53, 70, large)-net in base 27, because
- 15 times m-reduction [i] would yield (53, 55, large)-net in base 27, but