Best Known (81, 81+10, s)-Nets in Base 7
(81, 81+10, 2308331)-Net over F7 — Constructive and digital
Digital (81, 91, 2308331)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (12, 17, 2409)-net over F7, using
- net defined by OOA [i] based on linear OOA(717, 2409, F7, 6, 5) (dual of [(2409, 6), 14437, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(717, 2410, F7, 2, 5) (dual of [(2410, 2), 4803, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(73, 57, F7, 2, 2) (dual of [(57, 2), 111, 3]-NRT-code), using
- appending kth column [i] based on linear OA(73, 57, F7, 2) (dual of [57, 54, 3]-code), using
- Hamming code H(3,7) [i]
- appending kth column [i] based on linear OA(73, 57, F7, 2) (dual of [57, 54, 3]-code), using
- linear OOA(714, 2353, F7, 2, 5) (dual of [(2353, 2), 4692, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(714, 4706, F7, 5) (dual of [4706, 4692, 6]-code), using
- trace code [i] based on linear OA(497, 2353, F49, 5) (dual of [2353, 2346, 6]-code), using
- OOA 2-folding [i] based on linear OA(714, 4706, F7, 5) (dual of [4706, 4692, 6]-code), using
- linear OOA(73, 57, F7, 2, 2) (dual of [(57, 2), 111, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(717, 2410, F7, 2, 5) (dual of [(2410, 2), 4803, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(717, 2409, F7, 6, 5) (dual of [(2409, 6), 14437, 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 (12, 17, 2409)-net over F7, using
(81, 81+10, large)-Net over F7 — Digital
Digital (81, 91, large)-net over F7, using
- 71 times duplication [i] based on digital (80, 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
(81, 81+10, large)-Net in Base 7 — Upper bound on s
There is no (81, 91, large)-net in base 7, because
- 8 times m-reduction [i] would yield (81, 83, large)-net in base 7, but