Best Known (66, 66+20, s)-Nets in Base 7
(66, 66+20, 1681)-Net over F7 — Constructive and digital
Digital (66, 86, 1681)-net over F7, using
- net defined by OOA [i] based on linear OOA(786, 1681, F7, 20, 20) (dual of [(1681, 20), 33534, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(786, 16810, F7, 20) (dual of [16810, 16724, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(786, 16812, F7, 20) (dual of [16812, 16726, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [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(781, 16807, F7, 19) (dual of [16807, 16726, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(70, 5, F7, 0) (dual of [5, 5, 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(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(786, 16812, F7, 20) (dual of [16812, 16726, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(786, 16810, F7, 20) (dual of [16810, 16724, 21]-code), using
(66, 66+20, 12312)-Net over F7 — Digital
Digital (66, 86, 12312)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(786, 12312, F7, 20) (dual of [12312, 12226, 21]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(786, 16807, F7, 20) (dual of [16807, 16721, 21]-code), using
(66, 66+20, large)-Net in Base 7 — Upper bound on s
There is no (66, 86, large)-net in base 7, because
- 18 times m-reduction [i] would yield (66, 68, large)-net in base 7, but