Best Known (67, 67+11, s)-Nets in Base 7
(67, 67+11, 1152969)-Net over F7 — Constructive and digital
Digital (67, 78, 1152969)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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 (62, 73, 1152961)-net over F7, using
- net defined by OOA [i] based on linear OOA(773, 1152961, F7, 11, 11) (dual of [(1152961, 11), 12682498, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(773, 5764806, F7, 11) (dual of [5764806, 5764733, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(773, 5764809, F7, 11) (dual of [5764809, 5764736, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(773, 5764801, F7, 11) (dual of [5764801, 5764728, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(765, 5764801, F7, 10) (dual of [5764801, 5764736, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(773, 5764809, F7, 11) (dual of [5764809, 5764736, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(773, 5764806, F7, 11) (dual of [5764806, 5764733, 12]-code), using
- net defined by OOA [i] based on linear OOA(773, 1152961, F7, 11, 11) (dual of [(1152961, 11), 12682498, 12]-NRT-code), using
- digital (0, 5, 8)-net over F7, using
(67, 67+11, 5764831)-Net over F7 — Digital
Digital (67, 78, 5764831)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(778, 5764831, F7, 11) (dual of [5764831, 5764753, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(7) ⊂ Ce(5) [i] based on
- linear OA(773, 5764801, F7, 11) (dual of [5764801, 5764728, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(741, 5764801, F7, 6) (dual of [5764801, 5764760, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(73, 28, F7, 2) (dual of [28, 25, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- linear OA(71, 2, F7, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(10) ⊂ Ce(7) ⊂ Ce(5) [i] based on
(67, 67+11, large)-Net in Base 7 — Upper bound on s
There is no (67, 78, large)-net in base 7, because
- 9 times m-reduction [i] would yield (67, 69, large)-net in base 7, but