Regular Codes Are Not Asymptotically Good

In this note, we prove that the family of regular codes is not asymptotically good. The notation follows that in [2]. All codes considered are binary linear codes. H7 refers to the [7, 4] binary Hamming code. Definition 1. A binary linear code is regular iff it does not contain as a minor any code equivalent to H7 or H 7 . It follows from the theorem that the family of regular codes, which we will denote by R. Furthermore, R is closed under the taking of code duals, i.e., the dual of a regular code is also regular. This is because a code C contains H7 as a minor iff its dual C contains H 7 as a minor. It can further be shown [3, p. 437] that R is closed under the operations of direct sum, 2-sum, 3-sum and 3-sum; these operations are defined further below. Recall from coding theory that a code family C is called asymptotically good if there exists a sequence of [ni, ki, di] codes Ci ∈ C, with limi ni = ∞, such that lim infi ki/ni and lim infi di/ni are both strictly positive. Informally, in an asymptotically good code family, minimum distance and dimension can both grow linearly with the length of the code. The purpose of this note is to show the following theorem. Theorem 1. The family of regular codes is not asymptotically good. To prove Theorem 1, we need the following results. Theorem 2 ([6]). A code is graphic if and only if it does not contain as a minor any code equivalent to one of the codes H7, H 7 , C(K5) ⊥ and C(K3,3). Corollary 3. A code is cographic if and only if it does not contain as a minor any code equivalent to one of the codes H7, H 7 , C(K5) and C(K3,3). In the statement of the above theorem and corollary, K5 is the complete graph on five vertices, while K3,3 is the complete bipartite graph with three vertices on each side. C(K5) is the [10, 4, 4] code with generator matrix