Character prefix conditioning is an algorithm designed to improve code completion by enabling language models to sample token sequences based on a character prefix, rather than a token prefix, which addresses issues arising when a user's cursor is not on a token boundary. This approach is necessary because language models typically process sequences of tokens, and naive tokenization can produce incorrect results if the cursor is misaligned. The algorithm involves sampling sequences from a distribution defined by an autoregressive model, ensuring that the sequence starts with the specified character prefix. The challenge lies in constructing an efficient algorithm for sampling from this conditional distribution, minimizing calls to the language model, and the blog invites readers to contribute solutions.