On codes with local joint constraints

Linear Algebra and its Applications

2008

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This conjecture, known as the "finiteness conjecture", is now known to be false but no explicit counterexample to the conjecture is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that finite sets of nonnegative rational matrices have the finiteness property if and only if pairs of binary matrices do. We also show that all pairs of 2 × 2 binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for sets of nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices.

Section: Sets σmentioning

confidence: 99%

On the finiteness property for rational matrices

Linear Algebra and its Applications

2008

“…The case of binary matrices appears to be important in a number applications [8,[15][16][17][18]. These applications have led to a number of joint spectral radius computations for binary matrices [10,15,16].…”

Section: Theorem 1 the Finiteness Property Holds For All Sets Of Nonmentioning

confidence: 99%

On the Finiteness Property for Rational Matrices

The Joint Spectral Radius

2009

We analyze the periodicity of optimal long products of matrices. A set of matrices is said to have the finiteness property if the maximal rate of growth of long products of matrices taken from the set can be obtained by a periodic product. It was conjectured a decade ago that all finite sets of real matrices have the finiteness property. This "finiteness conjecture" is now known to be false but no explicit counterexample is available and in particular it is unclear if a counterexample is possible whose matrices have rational or binary entries. In this paper, we prove that all finite sets of nonnegative rational matrices have the finiteness property if and only if pairs of binary matrices do and we state a similar result when negative entries are allowed. We also show that all pairs of 2 × 2 binary matrices have the finiteness property. These results have direct implications for the stability problem for sets of matrices. Stability is algorithmically decidable for sets of matrices that have the finiteness property and so it follows from our results that if all pairs of binary matrices have the finiteness property then stability is decidable for nonnegative rational matrices. This would be in sharp contrast with the fact that the related problem of boundedness is known to be undecidable for sets of nonnegative rational matrices.

“…The easiest way of computing n is to apply (3), by evaluating the maximum-normed product of length n 0 m + 1 of matrices taken in the set 6. Moision et al mention in [13] an improvement of this brute force method: The main idea is to compute successively some sets of matrices 6 l , l = 1; 2 . .…”

Section: Upper and Lower Boundsmentioning

confidence: 99%

“…These are sets of products of length l, obtained by computing iteratively all products of a matrix in 6 l01 with a matrix in 6, and then removing from the set 6 l a matrix A, if it is dominated by another matrix B in this set, that is, if each entry of A is less or equal than the corresponding entry of B. For more information about this algorithm, we refer the reader to [13]. We propose here an improvement of this method: given the set 6 l , one can directly compute a set 6 2l by computing the set 6 2 l and then removing from this set all matrices that are dominated.…”

Section: Upper and Lower Boundsmentioning

confidence: 99%

On the Complexity of Computing the Capacity of Codes That Avoid Forbidden Difference Patterns

IEEE Trans. Inform. Theory

Protasov

2006

Abstract-Some questions related to the computation of the capacity of codes that avoid forbidden difference patterns are analysed. The maximal number of n-bit sequences whose pairwise differences do not contain some given forbidden difference patterns is known to increase exponentially with n; the coefficient of the exponent is the capacity of the forbidden patterns.In this paper, new inequalities for the capacity are given that allow for the approximation of the capacity with arbitrary high accuracy. The computational cost of the algorithm derived from these inequalities is fixed once the desired accuracy is given. Subsequently, a polynomial time algorithm is given for determining if the capacity of a set is positive while the same problem is shown to be NP-hard when the sets of forbidden patterns are defined over an extended set of symbols. Finally, the existence of extremal norms is proved for any set of matrices arising in the capacity computation. Based on this result, a second capacity approximating algorithm is proposed. The usefulness of this algorithm is illustrated by computing exactly the capacity of particular codes that were only known approximately.