Best Known (82, 82+9, s)-Nets in Base 7
(82, 82+9, 5764804)-Net over F7 — Constructive and digital
Digital (82, 91, 5764804)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (21, 25, 2882404)-net over F7, using
- net defined by OOA [i] based on linear OOA(725, 2882404, F7, 4, 4) (dual of [(2882404, 4), 11529591, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(725, 2882404, F7, 3, 4) (dual of [(2882404, 3), 8647187, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(725, 5764808, F7, 4) (dual of [5764808, 5764783, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(725, 5764809, F7, 4) (dual of [5764809, 5764784, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(725, 5764801, F7, 4) (dual of [5764801, 5764776, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(717, 5764801, F7, 3) (dual of [5764801, 5764784, 4]-code or 5764801-cap in PG(16,7)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(70, 8, F7, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(725, 5764809, F7, 4) (dual of [5764809, 5764784, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(725, 5764808, F7, 4) (dual of [5764808, 5764783, 5]-code), using
- appending kth column [i] based on linear OOA(725, 2882404, F7, 3, 4) (dual of [(2882404, 3), 8647187, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(725, 2882404, F7, 4, 4) (dual of [(2882404, 4), 11529591, 5]-NRT-code), using
- digital (57, 66, 2882402)-net over F7, using
- trace code for nets [i] based on digital (24, 33, 1441201)-net over F49, using
- net defined by OOA [i] based on linear OOA(4933, 1441201, F49, 9, 9) (dual of [(1441201, 9), 12970776, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(4933, 5764805, F49, 9) (dual of [5764805, 5764772, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- 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(4929, 5764801, F49, 8) (dual of [5764801, 5764772, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(4933, 5764805, F49, 9) (dual of [5764805, 5764772, 10]-code), using
- net defined by OOA [i] based on linear OOA(4933, 1441201, F49, 9, 9) (dual of [(1441201, 9), 12970776, 10]-NRT-code), using
- trace code for nets [i] based on digital (24, 33, 1441201)-net over F49, using
- digital (21, 25, 2882404)-net over F7, using
(82, 82+9, large)-Net over F7 — Digital
Digital (82, 91, large)-net over F7, using
- 71 times duplication [i] based on digital (81, 90, large)-net over F7, using
- t-expansion [i] based on digital (78, 90, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(790, large, F7, 12) (dual of [large, large−90, 13]-code), using
- trace code [i] based on linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- trace code [i] based on linear OA(4945, 5764801, F49, 12) (dual of [5764801, 5764756, 13]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(790, large, F7, 12) (dual of [large, large−90, 13]-code), using
- t-expansion [i] based on digital (78, 90, large)-net over F7, using
(82, 82+9, large)-Net in Base 7 — Upper bound on s
There is no (82, 91, large)-net in base 7, because
- 7 times m-reduction [i] would yield (82, 84, large)-net in base 7, but