Best Known (31, 42, s)-Nets in Base 27
(31, 42, 106290)-Net over F27 — Constructive and digital
Digital (31, 42, 106290)-net over F27, using
- net defined by OOA [i] based on linear OOA(2742, 106290, F27, 11, 11) (dual of [(106290, 11), 1169148, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2742, 531451, F27, 11) (dual of [531451, 531409, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 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(2733, 531442, F27, 9) (dual of [531442, 531409, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(2742, 531451, F27, 11) (dual of [531451, 531409, 12]-code), using
(31, 42, 528971)-Net over F27 — Digital
Digital (31, 42, 528971)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2742, 528971, F27, 11) (dual of [528971, 528929, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2742, 531451, F27, 11) (dual of [531451, 531409, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- 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(2733, 531442, F27, 9) (dual of [531442, 531409, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 278−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (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([0,5]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2742, 531451, F27, 11) (dual of [531451, 531409, 12]-code), using
(31, 42, large)-Net in Base 27 — Upper bound on s
There is no (31, 42, large)-net in base 27, because
- 9 times m-reduction [i] would yield (31, 33, large)-net in base 27, but