Codes for Updating Linear Functions over Small Fields

We consider a point-to-point communication scenario where the receiver intends to maintain a specific linear function of a message vector while the transmitter has access to an updated version of the message. The transmitter is required to broadcast a coded version of the updated message while the receiver must use this codeword and the current value of the linear function to update its contents. Under the assumption that the update is sparse and the transmitter does not know the exact value of the update vector, the objective is to design a linear code, with as small a codelength as possible, that allows successful update of the linear function at the receiver. This problem is motivated by applications to distributed data storage systems. A field-size independent lower bound on the codelength and a coding scheme meeting this bound were given by Prakash and Médard recently. However, this scheme requires a field size that grows quickly with the system parameters. In this paper, we provide a field-size aware analysis of the function update problem, including a tighter lower bound on the codelength, and design codes that allow us to trade-off the codelength for smaller field size requirements. Whenever the achieved codelengths equal those reported by Prakash and Médard the requirements on the size of the finite field are matched as well. Further, we identify the family of function update problems where linear coding provides reduction in codelength compared to a naive transmission scheme, and we also show that every function update problem is equivalent to a generalized index coding or functional index coding problem.