Best Known (80, 80+9, s)-Nets in Base 7
(80, 80+9, 4194350)-Net over F7 — Constructive and digital
Digital (80, 89, 4194350)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (3, 7, 50)-net over F7, using
- base reduction for projective spaces (embedding PG(3,49) in PG(6,7)) for nets [i] based on digital (0, 4, 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
- base reduction for projective spaces (embedding PG(3,49) in PG(6,7)) for nets [i] based on digital (0, 4, 50)-net over F49, using
- digital (73, 82, 4194300)-net over F7, using
- net defined by OOA [i] based on linear OOA(782, 4194300, F7, 10, 9) (dual of [(4194300, 10), 41942918, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(782, 8388601, F7, 2, 9) (dual of [(8388601, 2), 16777120, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(782, 8388602, F7, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
- trace code [i] based on linear OOA(4941, 4194301, F49, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4941, 8388602, F49, 9) (dual of [8388602, 8388561, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(4941, large, F49, 9) (dual of [large, large−41, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 28247525 | 4910−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(4941, large, F49, 9) (dual of [large, large−41, 10]-code), using
- OOA 2-folding [i] based on linear OA(4941, 8388602, F49, 9) (dual of [8388602, 8388561, 10]-code), using
- trace code [i] based on linear OOA(4941, 4194301, F49, 2, 9) (dual of [(4194301, 2), 8388561, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(782, 8388602, F7, 2, 9) (dual of [(8388602, 2), 16777122, 10]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(782, 8388601, F7, 2, 9) (dual of [(8388601, 2), 16777120, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(782, 4194300, F7, 10, 9) (dual of [(4194300, 10), 41942918, 10]-NRT-code), using
- digital (3, 7, 50)-net over F7, using
(80, 80+9, large)-Net over F7 — Digital
Digital (80, 89, large)-net over F7, using
- t-expansion [i] based on digital (78, 89, large)-net over F7, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on digital (78, 90, large)-net over F7, using
(80, 80+9, large)-Net in Base 7 — Upper bound on s
There is no (80, 89, large)-net in base 7, because
- 7 times m-reduction [i] would yield (80, 82, large)-net in base 7, but