Best Known (32−11, 32, s)-Nets in Base 27
(32−11, 32, 3938)-Net over F27 — Constructive and digital
Digital (21, 32, 3938)-net over F27, using
- net defined by OOA [i] based on linear OOA(2732, 3938, F27, 11, 11) (dual of [(3938, 11), 43286, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2732, 19691, F27, 11) (dual of [19691, 19659, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(2731, 19684, F27, 11) (dual of [19684, 19653, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(2725, 19684, F27, 9) (dual of [19684, 19659, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(2732, 19691, F27, 11) (dual of [19691, 19659, 12]-code), using
(32−11, 32, 13580)-Net over F27 — Digital
Digital (21, 32, 13580)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2732, 13580, F27, 11) (dual of [13580, 13548, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2732, 19691, F27, 11) (dual of [19691, 19659, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(2731, 19684, F27, 11) (dual of [19684, 19653, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(2725, 19684, F27, 9) (dual of [19684, 19659, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2732, 19691, F27, 11) (dual of [19691, 19659, 12]-code), using
(32−11, 32, large)-Net in Base 27 — Upper bound on s
There is no (21, 32, large)-net in base 27, because
- 9 times m-reduction [i] would yield (21, 23, large)-net in base 27, but