616 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-25, NO. 6, NOVEMBER 1979 On the Structure of Convolutional and Cyclic Convolutional Codes CORNELIS ROOS ,hrfac~--Algebraic convoiutionai coding theory is considered. It is shown tbat any convoiutionai code has a canonicai direct decomposition into s&code-s and tbat tbis decomposition leads in a natural way to a minimai encoder. Considering cyclic convolutional codes, as defined by Piret, an easy application of the generai theory yields a canonicai direct decomposition into cyclic subcodes, and at the same time a canonical minimai encoder for such codes. A list of pairs (n,k) admitting completely proper cyclic (n, k)-convolutional codes is included. I. INTRODUCTION T HIS PAPER presents a simple proof of the main structural result of Piret [6] on cyclic convolutional codes (CCC’s). Since a CCC is a left ideal in a semisimple algebra, it is decomposable as a direct sum of minimal left ideals. Piret shows that there exists one such decomposi- tion that leads in a natural way to the construction of a minimal encoder. This encoder is canonical from two points of view. As a minimal encoder (cf. Forney [3]) it minimizes the sum of the constraint lengths of its con- stituents, and at the same time it contains subencoders that each generate, as an F((D))-space, one of the above minimal ideals. However, Piret’s result is based on cumbersome matrix manipulations. Therefore, we shall present in this paper a more direct proof of the existence of the above canonical decomposition. The structure of the paper is as follows. Firstly we show, after having described the process of convolutional coding in Section II and having listed the appropriate definitions in Section III, that each convolu- tional code C has a canonical decomposition into sub- codes which induces in a natural way a minimal encoder of C (Section IV). In Section V we define C to be a CCC if it is an A-module, where A = F[x]mod(x” - l), gcd (IF 1, n) = 1. We apply well-known results on modules over semisimple rings in Section VI to produce a minimal encoder of C which reflects the cyclic structure of C optimally. We conclude by giving a nonexistence theorem for proper CCC’s in Section VII. This enables us to determine all pairs (n,k) admitting completely proper CCC’s, i.e., CCC’s containing no nonzero improper cyclic convolutional subcodes. Manuscript received October 19, 1977; revised March 15, 1979. T’he author is with the Department of Mathematics, University of Technology, Julianalaan 132, 2628 BL Delft, The Netherlands. Fig. 1. General convolutional encoder. II. CONVOLUTIONAL ENCODING A general convolutional encoder is shown in Fig. 1. At each unit of time the encoder accepts at the input a k-letter word, and an n-letter word is produced at the output. The letters are assumed to be taken from some finite field F. Using the delay operator D, we can repre- sent a sequence of k-letter words at the input as y(D)= 5 D’yj (1) j-o where yi is the k-letter word that enters the encoder at time j, and (I E Z. Similarly, the sequence of n-letter words produced by the encoder is denoted by cc v(D)= 2 D’vi, j=o . where q is the n-letter word that leaves the encoder at time j. The set of all sequences v(D) which are produced by the encoder when y(D) is allowed to run through all possible sequences of the form (1) is called the convolu- tional code generated by the encoder. Two encoders are called equivalent if they generate the same convolutional code. Following Forney [3] we shall restrict ourselves to en- coders which are constant linear causal finite-state sequential circuits. In that case any convolutional code can be characterized algebraically as a subspace of a suitably defined vector space. To clarify this let C be the convolutional code generatedby any such encoder, Let ri denote the response of the encoder to the unit k-letter word e,= (ei,, * - - ,e,) with e,, = Si, applied to the encoder at time 0, 1 =G i Gk. Now Forney has shown that by choos- ing the encoder appropriately 7i has the form 4 Ti= 2 Dj7.. cl' miEZ, m,>O, T~EF”. j-0 Furthermore, if we apply a shift of time to e,, let us say 0018-9448/79/ 1lOO-0676$00.75 0 1979IEEE