Best Known (60−15, 60, s)-Nets in Base 27
(60−15, 60, 75922)-Net over F27 — Constructive and digital
Digital (45, 60, 75922)-net over F27, using
- 271 times duplication [i] based on digital (44, 59, 75922)-net over F27, using
- net defined by OOA [i] based on linear OOA(2759, 75922, F27, 15, 15) (dual of [(75922, 15), 1138771, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2759, 531455, F27, 15) (dual of [531455, 531396, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(2757, 531441, F27, 15) (dual of [531441, 531384, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(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(14) ⊂ Ce(11) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(2759, 531455, F27, 15) (dual of [531455, 531396, 16]-code), using
- net defined by OOA [i] based on linear OOA(2759, 75922, F27, 15, 15) (dual of [(75922, 15), 1138771, 16]-NRT-code), using
(60−15, 60, 531461)-Net over F27 — Digital
Digital (45, 60, 531461)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2760, 531461, F27, 15) (dual of [531461, 531401, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(2757, 531442, F27, 15) (dual of [531442, 531385, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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(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([0,7]) ⊂ C([0,5]) [i] based on
(60−15, 60, large)-Net in Base 27 — Upper bound on s
There is no (45, 60, large)-net in base 27, because
- 13 times m-reduction [i] would yield (45, 47, large)-net in base 27, but