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Basic structural properties of tail-biting trellises are investigated. We start with rigorous definitions of various types of minimality for such trellises. We then show that biproper tail-biting trellises are not necessarily minimal, even though every minimal tail-biting trellis is biproper. We introduce the notion of linear trellises and prove, by example, that a minimal tail-biting trellis need not have any linearity properties whatsoever. We prove that a trellis is linear if and only if it factors into a product of elementary trellises. Using this result, we show how to construct, for any given linear code C, a tail-biting trellis that minimizes the product of state-space sizes among all possible linear tail-biting trellises. We also prove that every minimal linear tail-biting trellis for C arises from a certain n × n characteristic matrix, and show how to compute this matrix in time O(n2) from any basis for C.