In this study, we consider the data compression with side information available at both the encoder and the decoder. The information source is assigned to a variable-length code that does not have to satisfy the prefix-free constraints. We define several classes of codes whose codeword lengths and error probabilities satisfy worse-case criteria in terms of side-information. As a main result, we investigate the exact first-order asymptotics with second-order bounds scaled as Θ(√n) as blocklength n increases under the regime of nonvanishing error probabilities. To get this result, we also derive its one-shot bounds by employing the cutoff operation.

In this paper, we propose a lower bound for the minimum distance of [n,k] linear codes which are specified by generator matrices whose rows are k vectors of a given sequence of n linearly independent vectors over a finite field. The Feng-Rao bound and the order bound give the lower bounds for the minimum distance of the dual codes of the codes considered in this paper. We show that the proposed bound gives the true minimum distance for Reed-Solomon and Reed-Muller codes and exceeds the Goppa bound for some L-type algebraic geometry codes.