Best Known (81, 81+27, s)-Nets in Base 27
(81, 81+27, 40881)-Net over F27 — Constructive and digital
Digital (81, 108, 40881)-net over F27, using
- 271 times duplication [i] based on digital (80, 107, 40881)-net over F27, using
- net defined by OOA [i] based on linear OOA(27107, 40881, F27, 27, 27) (dual of [(40881, 27), 1103680, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(27107, 531454, F27, 27) (dual of [531454, 531347, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(27107, 531455, F27, 27) (dual of [531455, 531348, 28]-code), using
- 1 times truncation [i] based on linear OA(27108, 531456, F27, 28) (dual of [531456, 531348, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(23) [i] based on
- linear OA(27105, 531441, F27, 28) (dual of [531441, 531336, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(2793, 531441, F27, 24) (dual of [531441, 531348, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(273, 15, F27, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,27) or 15-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 Ce(27) ⊂ Ce(23) [i] based on
- 1 times truncation [i] based on linear OA(27108, 531456, F27, 28) (dual of [531456, 531348, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(27107, 531455, F27, 27) (dual of [531455, 531348, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(27107, 531454, F27, 27) (dual of [531454, 531347, 28]-code), using
- net defined by OOA [i] based on linear OOA(27107, 40881, F27, 27, 27) (dual of [(40881, 27), 1103680, 28]-NRT-code), using
(81, 81+27, 523455)-Net over F27 — Digital
Digital (81, 108, 523455)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27108, 523455, F27, 27) (dual of [523455, 523347, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(27108, 531461, F27, 27) (dual of [531461, 531353, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,11]) [i] based on
- linear OA(27105, 531442, F27, 27) (dual of [531442, 531337, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(2789, 531442, F27, 23) (dual of [531442, 531353, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (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([0,13]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(27108, 531461, F27, 27) (dual of [531461, 531353, 28]-code), using
(81, 81+27, large)-Net in Base 27 — Upper bound on s
There is no (81, 108, large)-net in base 27, because
- 25 times m-reduction [i] would yield (81, 83, large)-net in base 27, but