Summary Authentication codes (A‐codes) are a well studied technique to provide unconditionally secure authentication. An A‐code is defined by a map that associates a pair formed by a message and a key to a tag. A‐codes linear in the keys have been studied for application to distributed authentication schemes. In this paper, we address the dual question, which is the study of A‐codes that are linear in the messages. This is usually an undesired property, except in the context of network coding. Regarding these A‐codes, we derive some lower bounds on security parameters when key space is known. We also show a lower bound on key size when security parameter values are given (with some special properties) and construct some codes meeting the bound. We finally present a variant of these codes that authenticate multiple messages with the same key while preserving unconditional security. Copyright © 2013 John Wiley & Sons, Ltd.