Best Known (74, 74+10, s)-Nets in Base 7
(74, 74+10, 2306022)-Net over F7 — Constructive and digital
Digital (74, 84, 2306022)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (5, 10, 100)-net over F7, using
- trace code for nets [i] based on digital (0, 5, 50)-net over F49, using
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F49 with g(F) = 0 and N(F) ≥ 50, using
- the rational function field F49(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 49)-sequence over F49, using
- trace code for nets [i] based on digital (0, 5, 50)-net over F49, 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 (5, 10, 100)-net over F7, using
(74, 74+10, large)-Net over F7 — Digital
Digital (74, 84, large)-net over F7, using
- 72 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
(74, 74+10, large)-Net in Base 7 — Upper bound on s
There is no (74, 84, large)-net in base 7, because
- 8 times m-reduction [i] would yield (74, 76, large)-net in base 7, but