Best Known (87−10, 87, s)-Nets in Base 7
(87−10, 87, 2307098)-Net over F7 — Constructive and digital
Digital (77, 87, 2307098)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (8, 13, 1176)-net over F7, using
- net defined by OOA [i] based on linear OOA(713, 1176, F7, 5, 5) (dual of [(1176, 5), 5867, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(713, 2353, F7, 5) (dual of [2353, 2340, 6]-code), using
- net defined by OOA [i] based on linear OOA(713, 1176, F7, 5, 5) (dual of [(1176, 5), 5867, 6]-NRT-code), using
- digital (64, 74, 2305922)-net over F7, using
- trace code for nets [i] based on digital (27, 37, 1152961)-net over F49, using
- net defined by OOA [i] based on linear OOA(4937, 1152961, F49, 10, 10) (dual of [(1152961, 10), 11529573, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(4937, 5764805, F49, 10) (dual of [5764805, 5764768, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(4937, 5764801, F49, 10) (dual of [5764801, 5764764, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(4933, 5764801, F49, 9) (dual of [5764801, 5764768, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(4937, 5764805, F49, 10) (dual of [5764805, 5764768, 11]-code), using
- net defined by OOA [i] based on linear OOA(4937, 1152961, F49, 10, 10) (dual of [(1152961, 10), 11529573, 11]-NRT-code), using
- trace code for nets [i] based on digital (27, 37, 1152961)-net over F49, using
- digital (8, 13, 1176)-net over F7, using
(87−10, 87, large)-Net over F7 — Digital
Digital (77, 87, large)-net over F7, using
- 75 times duplication [i] based on digital (72, 82, large)-net over F7, using
- t-expansion [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
- t-expansion [i] based on digital (71, 82, large)-net over F7, using
(87−10, 87, large)-Net in Base 7 — Upper bound on s
There is no (77, 87, large)-net in base 7, because
- 8 times m-reduction [i] would yield (77, 79, large)-net in base 7, but