It was recently observed in , that in index coding, learning the coding matrix used by the server can pose privacy concerns: curious clients can extract information about the requests and side information of other clients. One approach to mitigate such concerns is the use of k-limited-access schemes , that restrict each client to learn only part of the index coding matrix, and in particular, at most k rows. These schemes transform a linear index coding matrix of rank T to an alternate one, such that each client needs to learn at most k of the coding matrix rows to decode its requested message. This paper analyzes k-limited-access schemes. First, a worst-case scenario, where the total number of clients n is 2T -1 is studied. For this case, a novel construction of the coding matrix is provided and shown to be order-optimal in the number of transmissions. Then, the case of a general n is considered and two different schemes are designed and analytically and numerically assessed in their performance. It is shown that these schemes perform better than the one designed for the case n = 2 T -1.