Best Known (105−13, 105, s)-Nets in Base 7
(105−13, 105, 1921614)-Net over F7 — Constructive and digital
Digital (92, 105, 1921614)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 13)-net over F7, using
- 6 times m-reduction [i] based on digital (1, 13, 13)-net over F7, using
- digital (85, 98, 1921601)-net over F7, using
- net defined by OOA [i] based on linear OOA(798, 1921601, F7, 14, 13) (dual of [(1921601, 14), 26902316, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(798, 5764804, F7, 2, 13) (dual of [(5764804, 2), 11529510, 14]-NRT-code), using
- trace code [i] based on linear OOA(4949, 2882402, F49, 2, 13) (dual of [(2882402, 2), 5764755, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4949, 5764804, F49, 13) (dual of [5764804, 5764755, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(4949, 5764805, F49, 13) (dual of [5764805, 5764756, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(4949, 5764801, F49, 13) (dual of [5764801, 5764752, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 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]
- 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(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(4949, 5764805, F49, 13) (dual of [5764805, 5764756, 14]-code), using
- OOA 2-folding [i] based on linear OA(4949, 5764804, F49, 13) (dual of [5764804, 5764755, 14]-code), using
- trace code [i] based on linear OOA(4949, 2882402, F49, 2, 13) (dual of [(2882402, 2), 5764755, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(798, 5764804, F7, 2, 13) (dual of [(5764804, 2), 11529510, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(798, 1921601, F7, 14, 13) (dual of [(1921601, 14), 26902316, 14]-NRT-code), using
- digital (1, 7, 13)-net over F7, using
(105−13, 105, large)-Net over F7 — Digital
Digital (92, 105, large)-net over F7, using
- 1 times m-reduction [i] based on digital (92, 106, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7106, large, F7, 14) (dual of [large, large−106, 15]-code), using
- trace code [i] based on linear OA(4953, 5764801, F49, 14) (dual of [5764801, 5764748, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- trace code [i] based on linear OA(4953, 5764801, F49, 14) (dual of [5764801, 5764748, 15]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7106, large, F7, 14) (dual of [large, large−106, 15]-code), using
(105−13, 105, large)-Net in Base 7 — Upper bound on s
There is no (92, 105, large)-net in base 7, because
- 11 times m-reduction [i] would yield (92, 94, large)-net in base 7, but