Best Known (61−9, 61, s)-Nets in Base 7
(61−9, 61, 1441210)-Net over F7 — Constructive and digital
Digital (52, 61, 1441210)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (48, 57, 1441202)-net over F7, using
- net defined by OOA [i] based on linear OOA(757, 1441202, F7, 9, 9) (dual of [(1441202, 9), 12970761, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(757, 5764809, F7, 9) (dual of [5764809, 5764752, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(757, 5764801, F7, 9) (dual of [5764801, 5764744, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(757, 5764809, F7, 9) (dual of [5764809, 5764752, 10]-code), using
- net defined by OOA [i] based on linear OOA(757, 1441202, F7, 9, 9) (dual of [(1441202, 9), 12970761, 10]-NRT-code), using
- digital (0, 4, 8)-net over F7, using
(61−9, 61, 5764829)-Net over F7 — Digital
Digital (52, 61, 5764829)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(761, 5764829, F7, 9) (dual of [5764829, 5764768, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- linear OA(757, 5764801, F7, 9) (dual of [5764801, 5764744, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(733, 5764801, F7, 5) (dual of [5764801, 5764768, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
(61−9, 61, large)-Net in Base 7 — Upper bound on s
There is no (52, 61, large)-net in base 7, because
- 7 times m-reduction [i] would yield (52, 54, large)-net in base 7, but