Best Known (65, 65+18, s)-Nets in Base 7
(65, 65+18, 1871)-Net over F7 — Constructive and digital
Digital (65, 83, 1871)-net over F7, using
- net defined by OOA [i] based on linear OOA(783, 1871, F7, 18, 18) (dual of [(1871, 18), 33595, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(783, 16839, F7, 18) (dual of [16839, 16756, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- 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(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(77, 32, F7, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- OA 9-folding and stacking [i] based on linear OA(783, 16839, F7, 18) (dual of [16839, 16756, 19]-code), using
(65, 65+18, 16839)-Net over F7 — Digital
Digital (65, 83, 16839)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(783, 16839, F7, 18) (dual of [16839, 16756, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- 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(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(77, 32, F7, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
(65, 65+18, large)-Net in Base 7 — Upper bound on s
There is no (65, 83, large)-net in base 7, because
- 16 times m-reduction [i] would yield (65, 67, large)-net in base 7, but