Best Known (42, 42+11, s)-Nets in Base 27
(42, 42+11, 1677720)-Net over F27 — Constructive and digital
Digital (42, 53, 1677720)-net over F27, using
- 272 times duplication [i] based on digital (40, 51, 1677720)-net over F27, using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2751, 8388601, F27, 11) (dual of [8388601, 8388550, 12]-code), using
- net defined by OOA [i] based on linear OOA(2751, 1677720, F27, 11, 11) (dual of [(1677720, 11), 18454869, 12]-NRT-code), using
(42, 42+11, large)-Net over F27 — Digital
Digital (42, 53, large)-net over F27, using
- 272 times duplication [i] based on digital (40, 51, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2751, large, F27, 11) (dual of [large, large−51, 12]-code), using
(42, 42+11, large)-Net in Base 27 — Upper bound on s
There is no (42, 53, large)-net in base 27, because
- 9 times m-reduction [i] would yield (42, 44, large)-net in base 27, but