Wash the limes with hot water and rub dry. Peel the lime peel very thinly. Squeeze 4 limes. Peel half of the ginger and cut into large pieces.
Put the lime zest with the pieces of ginger and the grain in a carafe and let it steep for at least 24 hours - better 48 hours.
Let the water with the rock sugar boil for about 10 minutes. Peel the remaining ginger and cut into large pieces. Add the honey, ginger and the lime juice to the sugar syrup. Let simmer for about 1/2 minute, then remove from heat and let cool down completely.
Pour the grain-lime mixture and the honey-sugar syrup through a coffee filter or a very fine hair sieve. Mix both in a carafe and keep closed in a cool place.