Best Known (68, 68+17, s)-Nets in Base 7
(68, 68+17, 14706)-Net over F7 — Constructive and digital
Digital (68, 85, 14706)-net over F7, using
- net defined by OOA [i] based on linear OOA(785, 14706, F7, 17, 17) (dual of [(14706, 17), 249917, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(785, 117649, F7, 17) (dual of [117649, 117564, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(785, 117649, F7, 17) (dual of [117649, 117564, 18]-code), using
(68, 68+17, 58827)-Net over F7 — Digital
Digital (68, 85, 58827)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(785, 58827, F7, 2, 17) (dual of [(58827, 2), 117569, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(785, 117654, F7, 17) (dual of [117654, 117569, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(785, 117655, F7, 17) (dual of [117655, 117570, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(785, 117649, F7, 17) (dual of [117649, 117564, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(779, 117649, F7, 16) (dual of [117649, 117570, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(70, 6, F7, 0) (dual of [6, 6, 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(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(785, 117655, F7, 17) (dual of [117655, 117570, 18]-code), using
- OOA 2-folding [i] based on linear OA(785, 117654, F7, 17) (dual of [117654, 117569, 18]-code), using
(68, 68+17, large)-Net in Base 7 — Upper bound on s
There is no (68, 85, large)-net in base 7, because
- 15 times m-reduction [i] would yield (68, 70, large)-net in base 7, but