The Berlekamp-Massey (1968, 1969) algorithm (BMA) and the Euclidean algorithm (EA) for decoding have been considered as two different algorithms for solving the same problem, namely, the one given by the key equation. We argue that they are essentially identical by showing how one can be adapted to perform the same arithmetic as the other. As a tool we use Feng and Tzeng's (1991) fundamental iterative algorithm that when adapted to the syndrome matrix is regarded as equivalent to the BMA.

Subgroup subcodes and their duals which can be considered as an extension of subfield subcodes and trace codes are defined. Some of their basic properties are investigated. Examples of good subgroup subcodes are included.>

A class of q/sup m/-ary constacyclic codes is defined which has the property that the q-ary image is equivalent to a class of shortened cyclic codes. This description leads to the construction of nonlinear codes. As an example one can construct nonlinear codes of length N=q/sup m/+1 over the field GF(q/sup m'/), m'/spl les/m, with a minimum distance greater than N(1-R)m'/m, where R is the rate. For higher rates these codes are the best known. The codes are easily encoded and decoded.>

The construction of arbitrarily long binary cyclic concatenated codes that are based on interleaved inner codes is described. The codes compare favorably with binary BCH codes of primitive length.>

The known construction of cyclic concatenated codes is based on the fact that the inner is a cyclic minimal code, the outer code is cyclic, and the lengths of the inner and outer codes are relatively prime. It is shown that if the outer code is a suitably chosen constacyclic code the overall concatenated code is always cyclic regardless of the length of the outer code. Moreover, it follows that any cyclic code of composite length is a direct sum of cyclic concatenated codes with inner cyclic minimal codes and outer constacyclic codes. This description of cyclic codes of composite length leads to the introduction of the concept of a poor-code length (PCL). All but low-rate codes of this length have a poor minimum distance. A PCL is directly related to the existence of irreducible binomials. In the binary case, the first few PCLs are 9, 25, 27, 45, 49, 75, 81 and 99. Arbitrarily long binary cyclic codes that are better than binary BCH codes of primitive length are constructed.>

Long binary sequences related to cyclic difference sets are investigated. Among all known constructions of cyclic difference sets it is shown that only sequences constructed from Hadamard difference sets can have an asymptotic nonzero merit factor. Maximal-length shift register sequences, Legendre, and twin-prime sequences are all constructed from Hadamard difference sets. The authors prove that the asymptotic merit factor of any maximal-length shift register sequence is three. For twin-prime sequences it is shown that the best asymptotic merit factor is six. This value is obtained by shifting the twin-prime sequence one quarter of its length. It turns out that Legendre sequences and twin-prime sequences have similar behavior. Jacobi sequences are investigated on the basis of the Jacobi symbol. The best asymptotic merit factor is shown to be six. Through the introduction of product sequences, it is argued that the maximal merit factor among all sequences of length N is at least six when N is large. The authors also demonstrate that it is fairly easy to construct sequences of moderate composite length with a merit factor close to six.>

Asymptotic bounds on the free distance of binary time-varying systematic convolutional codes are developed from a unified point of view. It is found that this unified approach provides some insight into the relationship between the distances of systematic versus nonsystematic codes. It is also helpful from a constructive point of view. The free-distance bounds of systematic unit memory codes are also discussed.>

In this paper a recursive algorithm using the error-trellis decoding technique is developed to decode certain convolutional codes (CC's). An example, illustrating the VLSI architecture of such a decoder, is given for a dual-k CC. It is demonstrated that such a decoder can be realized readily on a single chip with NMOS technology.

It is shown that a linear two-dimensional cyclic code of lengthnNcan be factorized into a direct sum of concatenated codes, with cyclic inner and outer codes and, conversely, thai a two-dimensional cyclic code can be constructed in this way. This result is extended, and it appears that the Abelian codes are obtainable by taking a direct sum of several concatenations of cyclic codes. Codes are constructed that are better than any previously known. In particular, low-rate cyclic codes superior to the duals of high-rate BCH codes are constructed.