DNA is an emerging medium for digital data and its adoption can be accelerated by synthesis processes specialized for storage applications. Here, we describe a de novo enzymatic synthesis strategy designed for data storage which harnesses the template-independent polymerase terminal deoxynucleotidyl transferase (TdT) in kinetically controlled conditions. Information is stored in transitions between non-identical nucleotides of DNA strands. To produce strands representing user-defined content, nucleotide substrates are added iteratively, yielding short homopolymeric extensions whose lengths are controlled by apyrase-mediated substrate degradation. With this scheme, we synthesize DNA strands carrying 144 bits, including addressing, and demonstrate retrieval with streaming nanopore sequencing. We further devise a digital codec to reduce requirements for synthesis accuracy and sequencing coverage, and experimentally show robust data retrieval from imperfectly synthesized strands. This work provides distributive enzymatic synthesis and information-theoretic approaches to advance digital information storage in DNA.