Best Known (49, 49+12, s)-Nets in Base 7
(49, 49+12, 19609)-Net over F7 — Constructive and digital
Digital (49, 61, 19609)-net over F7, using
- net defined by OOA [i] based on linear OOA(761, 19609, F7, 12, 12) (dual of [(19609, 12), 235247, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(761, 117654, F7, 12) (dual of [117654, 117593, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(761, 117655, F7, 12) (dual of [117655, 117594, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(755, 117649, F7, 11) (dual of [117649, 117594, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(761, 117655, F7, 12) (dual of [117655, 117594, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(761, 117654, F7, 12) (dual of [117654, 117593, 13]-code), using
(49, 49+12, 88794)-Net over F7 — Digital
Digital (49, 61, 88794)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(761, 88794, F7, 12) (dual of [88794, 88733, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using
- an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using
(49, 49+12, large)-Net in Base 7 — Upper bound on s
There is no (49, 61, large)-net in base 7, because
- 10 times m-reduction [i] would yield (49, 51, large)-net in base 7, but