In this paper, we show that a straightforward rewrite of a known minimal polynomial algorithm yields a simpler version of an algorithm in a recent paper of A. Salagean.

Let g be the genus of the Hermitian function field $H/{\mathbb F}_{q^2}$ and let $C_{\cal L}(D,mQ_{\infty})$ be a typical Hermitian code of length n. In [Des. Codes Cryptogr., to appear], we determined the dimension/length profile (DLP) lower bound on the state complexity of $C_{\cal L}(D,mQ_{\infty})$. Here we determine when this lower bound is tight and when it is not. For $m\leq \frac{n-2}{2}$ or $m\geq \frac{n-2}{2}+2g$, the DLP lower bounds reach Wolf's upper bound on state complexity and thus are trivially tight. We begin by showing that for about half of the remaining values of m the DLP bounds cannot be tight. In these cases, we give a lower bound on the absolute state complexity of $C_{\cal L}(D,mQ_{\infty})$, which improves the DLP lower bound. Next we give a "good" coordinate order for $C_{\cal L}(D,mQ_{\infty})$. With this good order, the state complexity of $C_{\cal L}(D,mQ_{\infty})$ achieves its DLP bound (whenever this is possible). This coordinate order also provides an upper bound on the absolute state complexity of $C_{\cal L}(D,mQ_{\infty})$ (for those values of m for which the DLP bounds cannot be tight). Our bounds on absolute state complexity do not meet for some of these values of m, and this leaves open the question whether our coordinate order is best possible in these cases. A straightforward application of these results is that if $C_{\cal L}(D,mQ_{\infty})$ is self-dual, then its state complexity (with respect to the lexicographic coordinate order) achieves its DLP bound of $\frac {n}{2}-\frac{q^2}{4}$, and, in particular, so does its absolute state complexity.

We study Reed-Muller codes and "Berman" codes as Abelian codes. We show that the duals of Berman codes and Reed-Muller codes can be considered as belonging to the same family of Abelian codes. We also determine the minimum distance and state complexity of the duals of Berman codes. Each of the classical parameters generalizes that of Reed-Muller codes in the obvious way, but the state complexity does not. We conclude by comparing the asymptotic behavior of the state complexity of the duals of Berman codes with that of the obvious generalization of the state complexity of Reed-Muller codes.

Let R be a finite chain ring (e.g., a Galois ring), K its residue field, and C a linear code over R. We prove that d(C), the Hamming distance of C, is d((~C~:~/spl alpha/~)~), where (C:/spl alpha/) is a submodule quotient, /spl alpha/ is a certain element of R, and denotes the canonical projection to K. These two codes also have the same set of minimal codeword supports. We explicitly construct a generator matrix/polynomial of (~C~:~/spl alpha/~)~ from the generator matrix/polynomials of C. We show that in general d(C)/spl les/d(C~) with equality for free codes (i.e., for free R-submodules of R/sup n/) and in particular for Hensel lifts of cyclic codes over K. Most of the codes over rings described in the literature fall into this class. We characterize minimum distance separable (MDS) codes over R and prove several analogs of properties of MDS codes over finite fields. We compute the Hamming weight enumerator of a free MDS code over R.

A new algebraic method to compute the transfer function of a trellis encoder with scalar labeling is outlined. This method is not difficult to implement and is found to be more efficient than conventional methods for encoders with a large number of states.>

The author introduce the n-dimensional key equation, which exhibits the error-locator polynomial of an n-dimensional cyclic code as a product of n univariate polynomials and the error-evaluator polynomial as an n-variable polynomial. They then reinterpret these polynomials in the context of linear recurring sequences. In particular, they reduce the decoding problem to successive application of the Berlekamp-Massey algorithm. With this new method, they are able to decode (up to half their minimum distance) many codes in a table of 2-D cyclic codes due to Jensen (1985).>

A sliding window decorrelating algorithm is developed. It is shown to be near-far resistant both in AWGN and fast fading environments. The algorithm is of particular relevance to base stations in multi-user DS-CDMA mobile radio networks, and is shown to alleviate the requirement for a complex power control algorithm to compensate for rapidly varying relative user energies.>

An n-dimensional linear recurring sequence ( sigma ) of elements from a field F, for n>or=1 is considered. An algorithm chi -BASE which determines a basis for the ideal characteristic polynomials of ( sigma ) under certain reasonable conditions is presented. This analysis applied to doubly periodic arrays in particular.>

The precise average and worst-case running times of the right- and left-shift gcd algorithms for $GF(q)[x]$ are derived. A new approximate integer model for the binary greatest common divisor algorithm is obtained. The right-shift polynomial worst case differs markedly for $q = 2$ and $q > 2$. The method also yields an easy analysis of the Euclidean algorithm.