A simple approach to recent generalizations of BCH and Goppa codes, viewed as subfield subcodes of modified Reed-Solomon codes, is presented. The orthogonal complements of these codes are characterized, and generalized versions of some previous results about minimum distance and decoding are obtained.

Periodically, some m of the n redundant components of a dependable system may have to be taken out of service for inspection, testing or preventive maintenance. The system is then constrained to operate with lower (n−m) redundancy and thus with less reliability during these periods. However, more frequent periodic inspections decrease the probability that a component fail undetected in the time interval between successive inspections. An optimal time schedule of periodic preventive operations arises from these two conflicting factors, balancing the loss of redundancy during inspections against the reliability benefits of more frequent inspections. Considering no other factor than this decreased redundancy at inspection time, this paper demonstrates the existence of an optimal interval between inspections, which maximizes the mean time between system failures. By suitable transformations and variable identifications, an analytic closed form expression of the optimum is obtained for the general (m, n) case. The optimum is shown to be unique within the ranges of parameter values valid in practice; its expression is easy to evaluate and shown to be useful to analyze and understand the influence of these parameters. Inspections are assumed to be perfect, i.e. they cause no component failure by themselves and leave no failure undetected. In this sense, the optimum determines a lowest bound for the system failure rate that can be achieved by a system of n-redundant components, m of which require for inspection or maintenance recurrent periods of unavailability of length t. The model and its general closed form solution are believed to be new [2], [5]. Previous work [1], [4], [10] had computed optimal values for an estimation of a time average of system unavailability, but by numerical procedures only and with different numerical approximations, other objectives and model assumptions (one component only inspected at a time), and taking into account failures caused by testing itself, repair and demands (see in particular [6], [7], [9]). System properties and practical implications are derived from the closed form analytical expression. Possible extensions of the model are discussed. The model has been applied to the scheduling of the periodic tests of nuclear reactor protection systems.

Consider a block code Γ of length n , over a q -ary alphabet. For a given integer t ∈ [0, n ], the regularity of Γ at level t is measured by the dimension of a well-defined subspace F of R t +1 , called the t-form space of the code. In particular, Γ is an orthogonal array of strength t if and only if F = R t +1 . A central result of the paper is the close connection that exists between t -forms and T-designs in the Hamming scheme H(n, q) . The t -form space F is shown to have the structure of an ideal, and the canonical generator of F to be computable from the distance distribution of the code Γ.

The Matsumoto-Imai public key scheme was developed to provide very fast signatures. It is based on substitution polynomials over GF (2 m ). This paper shows in two ways that the Matsumoto-Imai public key scheme is very easy to break. In the faster of the two attacks the time to cryptanalyze the scheme is about proportional to the binary length of the public key. This shows that Matsumoto and Imai greatly overestimated the security of their scheme.

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This paper contains a survey of association scheme theory (with its algebraic and analytical aspects) and of its applications to coding theory (in a wide sense). It is mainly concerned with a class of subjects that involve the central notion of the distance distribution of a code. Special emphasis is put on the linear programming method, inspired by the MacWilliams transform. This produces upper bounds for the size of a code with a given minimum distance, and lower bounds for the size of a design with a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certain well-defined “polynomial properties;” this leads one into the realm of orthogonal polynomial theory. In particular, some “universal bounds” are derived for codes and designs in polynomial type association schemes. Throughout the paper, the main concepts, methods, and results are illustrated by two examples that are of major significance in classical coding theory, namely, the Hamming scheme and the Johnson scheme. Other topics that receive special attention are spherical codes and designs, and additive codes in translation schemes, including Z4-additive binary codes

Given a code C, invariant linear forms are used to study the designs afforded by codewords of a fixed weight. The most important theorem relating codes and designs is due to Assmus and Mattson [J. Combin. Theory, 6 (1969), pp. 122–151], and this theorem is extended in different ways. For extremal self dual codes over the fields $\mathbb{F}_2 $ and $\mathbb{F}_3 $, it is proved that the t-designs afforded by the codewords of any fixed weight exhibit extra regularity with respect to $( t + 2 )$-sets. The same is true for the design afforded by the codewords of minimum weight in an extremal self-dual code over $\mathbb{F}_4 $. The invariant linear forms are also used to construct Boolean designs with several block sizes, extending previous work by Safavi-Naini and Blake [Utilitas Math., 14 (1978), pp. 49–63], [Ars Combin., 7 (1979), pp. 135–151], [Inform. and Control, 42 (1986), pp. 261–282].

Predictor polynomials corresponding to nested Toeplitz matrices are known to be connected by the Szego-Levinson recurrence relation. A generalization of that result, where the relevant reduction process for Toeplitz matrices (of decreasing order) is defined by an elementary one-parameter linear transformation, is addressed. The descending and ascending versions of the corresponding generalized Szego-Levinson recurrence relations are discussed in detail. In particular, these relations are shown to be essentially the same as the extraction formulas for canonical Schur and Brune sections in the Dewilde-Dym (1984) recursive solution of the lossless inverse scattering problem. Some extensions of the Levinson algorithm for linear prediction and of the Schur-Cohn algorithm for polynomial stability test are presented.< >

The central subject of this paper is the three-term recurrence formula satisfied by the symmetric (first-kind) and antisymmetric (second-kind) polynomials relative to a given sequence of reflection coefficients, the last element of which has unit modulus. The theory of these polynomials is shown to have interesting analogies with the classical theory of orthogonal polynomials on the real line. In the case of real data, the former is equivalent to a special case of the latter (by a change of variable). The main application considered here is the problem of computing the zeros of the highest-degree symmetric polynomial, which is identified as a predictor polynomial. This problem occurs not only in some modelling techniques for digital signal processing, but can also be interpreted as the eigenvalue problem for a unitary Hessenberg matrix. Attractive solution methods are derived from the "tridiagonal approach," based on the three-term recurrence relation.

This paper shows that the “split approach” produces new efficient algorithms to solve not only the Toeplitz linear prediction problem, as explained in a companion paper, but also a collection of other important DSP problems belonging to the Toeplitz environment. It emphasizes both the algorithmic procedures themselves, and their connections with standard mathematical subjects such as the theory of orthogonal polynomials and the theory of positive functions.

The split Levinson algorithms constitute a new class of efficient procedures to solve a linear system of equations exhibiting the positive definite Toeplitz structure. They can be derived from the classical Levinson algorithm by a kind of splitting operation, which results in a more efficient algorithm processing some well-defined symmetric polynomials (with complex coefficients). The algorithm thus obtained is based on a simple two-step recurrence relation satisfied by these polynomials. The paper provides a self-contained introduction to the whole class of split Levinson algorithms, including a detailed technical derivation. This class is shown to depend on two unit modulus parameters, which can be chosen at will by the user.

Several authors have pointed out some limitations inherent in Montague's approach to natural language representation, and have attempted to remedy these limitations by working out suitable extensions of the Montague formalism. One of these limitations originates from the fact that logical connectives and modal operators can only be applied to formulae and not to arbitrary logical expressions, which stands in contradiction with the common practice of natural language where connectives and modalities can affect expressions of various syntactic categories (and not only complete sentences). This paper introduces an extension of Montague's intensional logic that precisely addresses the problem just alluded to. The proposed extension is equipped with a well-defined 'Boolean semantics', analogous with the Keenan-Faltz semantics.

The authors deal with the problem of verifying whether a given antisymmetric function is lossless (in the discrete sense), which is closely related to the polynomial stability problem. The proposed approach is based on some simple correspondences between the class of discrete-type lossless functions and the subclass of continuous-type lossless functions having all their finite poles and zeros in a fixed interval of the imaginary axis. They make it possible to derive a collection of discrete-type losslessness tests by a mere translation of suitable refinements of the classical Cauer criteria. The underlying recurrence relations are shown to have interesting interpretations in the classical framework of real line orthogonal polynomial theory

This paper contains a thorough investigation of a family of symmetric "predictor polynomials" associated with a nonnegative-definite Toeplitz matrix. These polynomials are constructed from the classical predictors and from the values assumed by some dual predictors in a fixed point of unit modulus; the appropriate duality is induced by changing the sequence of reflection coefficients into its conjugate mirror image, within a unit modulus factor. The central theme of the paper is a well-defined three-term recurrence relation satisfied by these symmetric polynomials; it motivates the "tridiagonal" terminology. The properties of the recurrence are studied in detail; special attention is paid to the important issue of computing the recurrence coefficients from the reflection coefficients. It is shown how this three-term recurrence formula produces an efficient solution method, called the split Levinson algorithm, for the linear prediction problem.

Let w/sub 1/=d,w/sub 2/,...,w/sub s/ be the weights of the nonzero codewords in a binary linear (n,k,d) code C, and let w'/sub 1/, w'/sub 2/, ..., w'/sub 3/, be the nonzero weights in the dual code C1. Let t be an integer in the range 0 or=d+4 then either the words of any nonzero weight w/sub i/ form a (t+1)-design or else the codewords of minimal weight d form a (1,2,...,t,t+2)-design. If in addition C is self-dual with all weights divisible by 4 then the codewords of any given weight w/sub i/ form either a (t +1)-design or a (1,2,...,t,t+2)-design. The proof avoids the use of modular forms. >

This paper is concerned with a novel three-term recurrence relation for a well-defined family of symmetric or antisymmetrie polynomials. In the context where it was originally discovered, this recurrence yields a method for checking whether a given polynomial is devoid of zeros in the closed unit disc. As shown in this paper, the same recurrence, used in reverse order, gives rise to a new algorithm for solving the classicallinear prediction problem relative to a given positive definite Toeplitz matrix. This algorithm can be viewed as a split version of a suitable analogue of the well-known Levinson algorithm.

This paper is devoted to a family of interpolation type problems for positive trigonometric polynomials of a given ordern. Via the Riesz-Fejér factorization theorem, they can be viewed as natural generalizations of the partial autocorrelation problem for discrete time signals of lengthn+1. The relevant variables for a specific problem are well-defined linear combinations of the coefficients of the underlying trigonometric polynomial. An efficient method is obtained to characterize the feasibility region of the problem, defined as the set of points having these variables as coordinates. It allows us to determine the boundary of that region by computing the extreme eigen values and the corresponding eigenvectors of certain well-defined Hermitian Toeplitz matrices of ordern+1. The method is an extension of one proposed by Steinhardt to solve the coefficient problem for positive cosine polynomials (which belongs to the family). Other interesting applications are the Nevanlinna-Pick interpolation problem for polynomial functions, and the simple interpolation problem for positive trigonometric polynomials. The close connection between the generalized Steinhardt method and classical techniques based on the polarity theorem for convex cones and on the Hahn-Banach extension theorem are established.