Best Known (51−13, 51, s)-Nets in Base 27
(51−13, 51, 88575)-Net over F27 — Constructive and digital
Digital (38, 51, 88575)-net over F27, using
- 271 times duplication [i] based on digital (37, 50, 88575)-net over F27, using
- net defined by OOA [i] based on linear OOA(2750, 88575, F27, 13, 13) (dual of [(88575, 13), 1151425, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2750, 531451, F27, 13) (dual of [531451, 531401, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,5]) [i] based on
- 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(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(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([0,6]) ⊂ C([0,5]) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(2750, 531451, F27, 13) (dual of [531451, 531401, 14]-code), using
- net defined by OOA [i] based on linear OOA(2750, 88575, F27, 13, 13) (dual of [(88575, 13), 1151425, 14]-NRT-code), using
(51−13, 51, 531455)-Net over F27 — Digital
Digital (38, 51, 531455)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2751, 531455, F27, 13) (dual of [531455, 531404, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(2749, 531441, F27, 13) (dual of [531441, 531392, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2737, 531441, F27, 10) (dual of [531441, 531404, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(12) ⊂ Ce(9) [i] based on
(51−13, 51, large)-Net in Base 27 — Upper bound on s
There is no (38, 51, large)-net in base 27, because
- 11 times m-reduction [i] would yield (38, 40, large)-net in base 27, but