Best Known (68−17, 68, s)-Nets in Base 27
(68−17, 68, 66432)-Net over F27 — Constructive and digital
Digital (51, 68, 66432)-net over F27, using
- net defined by OOA [i] based on linear OOA(2768, 66432, F27, 17, 17) (dual of [(66432, 17), 1129276, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2768, 531457, F27, 17) (dual of [531457, 531389, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2768, 531461, F27, 17) (dual of [531461, 531393, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [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(2749, 531442, F27, 13) (dual of [531442, 531393, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,8]) ⊂ C([0,6]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2768, 531461, F27, 17) (dual of [531461, 531393, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2768, 531457, F27, 17) (dual of [531457, 531389, 18]-code), using
(68−17, 68, 531461)-Net over F27 — Digital
Digital (51, 68, 531461)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2768, 531461, F27, 17) (dual of [531461, 531393, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [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(2749, 531442, F27, 13) (dual of [531442, 531393, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,8]) ⊂ C([0,6]) [i] based on
(68−17, 68, large)-Net in Base 27 — Upper bound on s
There is no (51, 68, large)-net in base 27, because
- 15 times m-reduction [i] would yield (51, 53, large)-net in base 27, but