Best Known (22, 33, s)-Nets in Base 27
(22, 33, 3938)-Net over F27 — Constructive and digital
Digital (22, 33, 3938)-net over F27, using
- 271 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(2732, 3938, F27, 11, 11) (dual of [(3938, 11), 43286, 12]-NRT-code), using
(22, 33, 19588)-Net over F27 — Digital
Digital (22, 33, 19588)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2733, 19588, F27, 11) (dual of [19588, 19555, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2733, 19694, F27, 11) (dual of [19694, 19661, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(2731, 19683, F27, 11) (dual of [19683, 19652, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(2722, 19683, F27, 8) (dual of [19683, 19661, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-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(10) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(2733, 19694, F27, 11) (dual of [19694, 19661, 12]-code), using
(22, 33, large)-Net in Base 27 — Upper bound on s
There is no (22, 33, large)-net in base 27, because
- 9 times m-reduction [i] would yield (22, 24, large)-net in base 27, but