Best Known (59, 69, s)-Nets in Base 7
(59, 69, 1152965)-Net over F7 — Constructive and digital
Digital (59, 69, 1152965)-net over F7, using
- net defined by OOA [i] based on linear OOA(769, 1152965, F7, 10, 10) (dual of [(1152965, 10), 11529581, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(769, 5764825, F7, 10) (dual of [5764825, 5764756, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(769, 5764829, F7, 10) (dual of [5764829, 5764760, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- 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(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(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(769, 5764829, F7, 10) (dual of [5764829, 5764760, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(769, 5764825, F7, 10) (dual of [5764825, 5764756, 11]-code), using
(59, 69, 5764829)-Net over F7 — Digital
Digital (59, 69, 5764829)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(769, 5764829, F7, 10) (dual of [5764829, 5764760, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- 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(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(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
(59, 69, large)-Net in Base 7 — Upper bound on s
There is no (59, 69, large)-net in base 7, because
- 8 times m-reduction [i] would yield (59, 61, large)-net in base 7, but