Best Known (69, 89, s)-Nets in Base 7
(69, 89, 1682)-Net over F7 — Constructive and digital
Digital (69, 89, 1682)-net over F7, using
- 71 times duplication [i] based on digital (68, 88, 1682)-net over F7, using
- net defined by OOA [i] based on linear OOA(788, 1682, F7, 20, 20) (dual of [(1682, 20), 33552, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(788, 16820, F7, 20) (dual of [16820, 16732, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- linear OA(786, 16807, F7, 20) (dual of [16807, 16721, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(776, 16807, F7, 18) (dual of [16807, 16731, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(771, 16807, F7, 17) (dual of [16807, 16736, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(71, 12, F7, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 342 = 73−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(71, 342, F7, 1) (dual of [342, 341, 2]-code), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(19) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- OA 10-folding and stacking [i] based on linear OA(788, 16820, F7, 20) (dual of [16820, 16732, 21]-code), using
- net defined by OOA [i] based on linear OOA(788, 1682, F7, 20, 20) (dual of [(1682, 20), 33552, 21]-NRT-code), using
(69, 89, 16825)-Net over F7 — Digital
Digital (69, 89, 16825)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(789, 16825, F7, 20) (dual of [16825, 16736, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(786, 16807, F7, 20) (dual of [16807, 16721, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(771, 16807, F7, 17) (dual of [16807, 16736, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(73, 18, F7, 2) (dual of [18, 15, 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(19) ⊂ Ce(16) [i] based on
(69, 89, large)-Net in Base 7 — Upper bound on s
There is no (69, 89, large)-net in base 7, because
- 18 times m-reduction [i] would yield (69, 71, large)-net in base 7, but