Best Known (53, 53+27, s)-Nets in Base 27
(53, 53+27, 1514)-Net over F27 — Constructive and digital
Digital (53, 80, 1514)-net over F27, using
- 271 times duplication [i] based on digital (52, 79, 1514)-net over F27, using
- net defined by OOA [i] based on linear OOA(2779, 1514, F27, 27, 27) (dual of [(1514, 27), 40799, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2779, 19683, F27, 27) (dual of [19683, 19604, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2779, 19684, F27, 27) (dual of [19684, 19605, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2779, 19684, F27, 27) (dual of [19684, 19605, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2779, 19683, F27, 27) (dual of [19683, 19604, 28]-code), using
- net defined by OOA [i] based on linear OOA(2779, 1514, F27, 27, 27) (dual of [(1514, 27), 40799, 28]-NRT-code), using
(53, 53+27, 13042)-Net over F27 — Digital
Digital (53, 80, 13042)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2780, 13042, F27, 27) (dual of [13042, 12962, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2780, 19691, F27, 27) (dual of [19691, 19611, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(2779, 19684, F27, 27) (dual of [19684, 19605, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(2773, 19684, F27, 25) (dual of [19684, 19611, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 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([0,13]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2780, 19691, F27, 27) (dual of [19691, 19611, 28]-code), using
(53, 53+27, large)-Net in Base 27 — Upper bound on s
There is no (53, 80, large)-net in base 27, because
- 25 times m-reduction [i] would yield (53, 55, large)-net in base 27, but