Best Known (56, 56+19, s)-Nets in Base 27
(56, 56+19, 59050)-Net over F27 — Constructive and digital
Digital (56, 75, 59050)-net over F27, using
- 271 times duplication [i] based on digital (55, 74, 59050)-net over F27, using
- net defined by OOA [i] based on linear OOA(2774, 59050, F27, 19, 19) (dual of [(59050, 19), 1121876, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2774, 531451, F27, 19) (dual of [531451, 531377, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(2773, 531442, F27, 19) (dual of [531442, 531369, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (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(271, 9, F27, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(2774, 531451, F27, 19) (dual of [531451, 531377, 20]-code), using
- net defined by OOA [i] based on linear OOA(2774, 59050, F27, 19, 19) (dual of [(59050, 19), 1121876, 20]-NRT-code), using
(56, 56+19, 469469)-Net over F27 — Digital
Digital (56, 75, 469469)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2775, 469469, F27, 19) (dual of [469469, 469394, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2775, 531455, F27, 19) (dual of [531455, 531380, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(2773, 531441, F27, 19) (dual of [531441, 531368, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(272, 14, F27, 2) (dual of [14, 12, 3]-code or 14-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2775, 531455, F27, 19) (dual of [531455, 531380, 20]-code), using
(56, 56+19, large)-Net in Base 27 — Upper bound on s
There is no (56, 75, large)-net in base 27, because
- 17 times m-reduction [i] would yield (56, 58, large)-net in base 27, but