Best Known (69−11, 69, s)-Nets in Base 7
(69−11, 69, 164717)-Net over F7 — Constructive and digital
Digital (58, 69, 164717)-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 (53, 64, 164709)-net over F7, using
- net defined by OOA [i] based on linear OOA(764, 164709, F7, 11, 11) (dual of [(164709, 11), 1811735, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(764, 823546, F7, 11) (dual of [823546, 823482, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(764, 823550, F7, 11) (dual of [823550, 823486, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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(764, 823550, F7, 11) (dual of [823550, 823486, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(764, 823546, F7, 11) (dual of [823546, 823482, 12]-code), using
- net defined by OOA [i] based on linear OOA(764, 164709, F7, 11, 11) (dual of [(164709, 11), 1811735, 12]-NRT-code), using
- digital (0, 5, 8)-net over F7, using
(69−11, 69, 823571)-Net over F7 — Digital
Digital (58, 69, 823571)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(769, 823571, F7, 11) (dual of [823571, 823502, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(767, 823567, F7, 11) (dual of [823567, 823500, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(73, 24, F7, 2) (dual of [24, 21, 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
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(767, 823569, F7, 10) (dual of [823569, 823502, 11]-code), using Gilbert–Varšamov bound and bm = 767 > Vbs−1(k−1) = 4 840332 546789 490830 563781 026182 561747 581262 573858 747777 [i]
- linear OA(70, 2, F7, 0) (dual of [2, 2, 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
- linear OA(767, 823567, F7, 11) (dual of [823567, 823500, 12]-code), using
- construction X with Varšamov bound [i] based on
(69−11, 69, large)-Net in Base 7 — Upper bound on s
There is no (58, 69, large)-net in base 7, because
- 9 times m-reduction [i] would yield (58, 60, large)-net in base 7, but