Despite much progress over the past decade, current Single Nucleotide Polymorphism (SNP) genotyping technologies still offer an insufficient degree of multiplexing when required to handle user-selected sets of SNPs. In this paper we propose a new genotyping assay architecture combining multiplexed solution-phase single-base extension (SBE) reactions with sequencing by hybridization (SBH) using universal DNA arrays such as all $k$-mer arrays. In addition to PCR amplification of genomic DNA, SNP genotyping using SBE/SBH assays involves the following steps: (1) Synthesizing primers complementing the genomic sequence immediately preceding SNPs of interest; (2) Hybridizing these primers with the genomic DNA; (3) Extending each primer by a single base using polymerase enzyme and dideoxynucleotides labeled with 4 different fluorescent dyes; and finally (4) Hybridizing extended primers to a universal DNA array and determining the identity of the bases that extend each primer by hybridization pattern analysis. Our contributions include a study of multiplexing algorithms for SBE/SBH genotyping assays and preliminary experimental results showing the achievable tradeoffs between the number of array probes and primer length on one hand and the number of SNPs that can be assayed simultaneously on the other. Simulation results on datasets both randomly generated and extracted from the NCBI dbSNP database suggest that the SBE/SBH architecture provides a flexible and cost-effective alternative to genotyping assays currently used in the industry, enabling genotyping of up to hundreds of thousands of user-specified SNPs per assay.