An Encounter with Convolutional Codes over Rings
Abstract: Convolutional codes is one possibility when there is a need for error-correcting codes in communication systems. Using convolutional codes over rings is a relatively new approach. When coding is used in combination with, for example, phase-shift keying, codes over rings constitue a natural choice as the symbols in the code alphabet have a natural signal point interpretation. This thesis reports on fundamental questions regarding convolutional codes over rings. A careful definition of convolutional codes over rings is presented. It is interesting to note that properties as systematicity, right invertibility, and minimality of generator matrices are code properties. In the case of convolutional codes over fields, these are properties connected to the chosen generator matrix only. The choice of generator matrix for a specific convolutional code is important for the behavior of the code. Structural properties as minimality, systematicity, the predictable degree property, right invertibility, catastrophicity, and basic and minimal-basic generator matrices are studied and reported on in the thesis. The direct sum decomposition of rings that satisfy the descending chain condition is used to further study generator matrix properties and code properties. Code search results for rate-1/2 convolutional codes over the ring of integers modulo 4 up to memory m=5 have been conducted. The obtained codes are compared with rate-1/2 convolutional codes over the binary field. Furthermore, an algorithm for constructing the code state trellis of convolutional codes over rings starting with an arbitrary generator matrix is presented.
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