Best Known (85−11, 85, s)-Nets in Base 7
(85−11, 85, 2305924)-Net over F7 — Constructive and digital
Digital (74, 85, 2305924)-net over F7, using
- 71 times duplication [i] based on digital (73, 84, 2305924)-net over F7, using
- trace code for nets [i] based on digital (31, 42, 1152962)-net over F49, using
- net defined by OOA [i] based on linear OOA(4942, 1152962, F49, 11, 11) (dual of [(1152962, 11), 12682540, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(4942, 5764811, F49, 11) (dual of [5764811, 5764769, 12]-code), using
- construction X applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(4941, 5764802, F49, 11) (dual of [5764802, 5764761, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(4933, 5764802, F49, 9) (dual of [5764802, 5764769, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 5764802 | 498−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(491, 9, F49, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(491, s, F49, 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(4942, 5764811, F49, 11) (dual of [5764811, 5764769, 12]-code), using
- net defined by OOA [i] based on linear OOA(4942, 1152962, F49, 11, 11) (dual of [(1152962, 11), 12682540, 12]-NRT-code), using
- trace code for nets [i] based on digital (31, 42, 1152962)-net over F49, using
(85−11, 85, large)-Net over F7 — Digital
Digital (74, 85, large)-net over F7, using
- 73 times duplication [i] based on digital (71, 82, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(782, large, F7, 11) (dual of [large, large−82, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176803 | 79−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(782, large, F7, 11) (dual of [large, large−82, 12]-code), using
(85−11, 85, large)-Net in Base 7 — Upper bound on s
There is no (74, 85, large)-net in base 7, because
- 9 times m-reduction [i] would yield (74, 76, large)-net in base 7, but