We consider a cache network consisting of storage-enabled servers forming a distributed content delivery scenario. Each incoming content request, arriving uniformly at random to these servers, will be redirected to one or more responsible servers for that request. While assigning each request to its nearest available replica seems to be a reasonable solution, specifically in terms of communication cost, it will result in overloaded servers. Thus, alternatively, we investigate a coded scheme which will address this problem. In this scheme, coded chunks of original files are stored in servers based on the files popularity distribution, which we consider to be Zipf. Then, upon each request arrival, by delivering enough coded chunks to the request origin, the request can be decoded.Specifically, we show that if n requests are assigned to n servers based on the proposed coded scheme, the maximum load of servers will be Θ(1), while in the nearest replica strategy the maximum load is Θ(log n). More surprisingly, this coded scheme will have the same communication cost performance as the nearest replica strategy, asymptotically. Finally, our numerical results show that the coded scheme surpasses the uncoded one even in non-asymptotic regimes.