On linear codes correcting random and low-density burst errors

This paper presents lower and upper bounds on the number of parity-check digits required for a linear code that corrects random errors and errors which are in the form of low-density bursts.     

On the weight of numerical semigroups

We investigate the weights of a family of numerical semigroups by means of even gaps and the Weierstrass property for such a family.     

On numerical semigroups and the order bound

Let S={s₀=0<s₁<?<s_i…}⊆N be a numerical non-ordinary semigroup; then set, for each . We find a non-negative integer m such that d_ORD(i)=ν_i+1 for i≥m, where d_ORD(i) denotes the order bound on the minimum distance of an algebraic geometry code associated to S. In several cases (including the acute ones, that have previously come up in the literature) we show that this integer m is the smallest one with the above property. Furthermore it is shown that every semigroup generated by an arithmetic sequence or generated by three elements is acute. For these semigroups, the value of m is also found.     

On comparing two chains of numerical semigroups and detecting arf semigroups

If T is a numerical semigroup with maximal ideal N , define associated semigroups B(T):=(N-N) and L(T)= \cup { (hN-hN) \colon h \geq 1 } . If S is a numerical semigroup, define strictly increasing finite sequences { B _i (S) \colon 0 ≤ i ≤β (S) } and { L _i (S) \colon 0 ≤ i ≤λ (S) } of semigroups by B ₀ (S):=S=:L ₀ (S) , B _{β (S)} (S):= \Bbb N =: L _{λ (S)} (S) , B _i+1 (S):=B(B _i (S)) for 0<i< β (S) , L _i+1 (S):=L(L _i (S)) for 0<i< λ (S) . It is shown, contrary to recent claims and conjectures, that B ₂ (S) need not be a subset of L ₂ (S) and that β (S) - λ (S) can be any preassigned integer. On the other hand, B ₂ (S) \subseteq L ₂ (S) in each of the following cases: S is symmetric;S has maximal embedding dimension;S has embedding dimension e(S) ≤ 3 . Moreover, if either e(S)=2 or S is pseudo-symmetric of maximal embedding dimension, then B _i (S) \subseteq L _i (S) for each i , 0 ≤ i ≤λ (S) . For each integer n \geq 2 , an example is given of a (necessarily non-Arf) semigroup S such that β (S) = λ (S)=n , B _i (S) = L _i (S) for all 0 ≤ i ≤ n-2 , and B _n-1 (S) \subsetneqq L _n-1 (S) . April 4, 2000     

On the variety of linear recurrences and numerical semigroups

In this work, we prove the existence of linear recurrences of order M with a non-trivial solution vanishing exactly on the set of gaps (or a subset) of a numerical semigroup S finitely generated by a ₁<a ₂<?<a _N and M=a _N .     

On numerical semigroups related to covering of curves

We investigate arithmetical properties of a class of semigroups that includesthose appearing as Weierstrass semigroups at totally ramified points of coveringof curves.     

On certain codes admitting inverse semigroups as syntactic monoids

The purpose of this paper is to investigate under what conditions an inverse semigroup M is isomorphic to the syntactic monoid M(A)* of afinite prefix code A over an alphabet X. We find a necessary condition for this to happen. It expresses a precise link between the group of units of M and the maximal subgroups of the 0-minimal ideal of M (Theorem 2.1). The condition is shown to be sufficient in case M is an ideal extension of a Brandt semigroup by a group (Corollary 2.3). We also introduce and study stable codes (products of subsets of the alphabet) and give structural properties of their syntactic monoids (Proposition 3.3 and Theorem 3.5). Most of our results inter-relate structural properties of certain semigroups and divisibility of integers attached to them. The terminology follows [1] and [3].     

On the second Feng-Rao distance of Algebraic Geometry codes related to Arf semigroups

We describe the second (generalized) Feng-Rao distance for elements in an Arf numerical semigroup that are greater than or equal to the conductor of the semigroup. This provides a lower bound for the second Hamming weight for one point AG codes. In particular, we can obtain the second Feng-Rao distance for the codes defined by asymptotically good towers of function fields whose Weierstrass semigroups are inductive. In addition, we compute the second Feng-Rao number, and provide some examples and comparisons with previous results on this topic. These calculations rely on Apéry sets, and thus several results concerning Apéry sets of Arf semigroups are presented.     

On average case errors in numerical analysis

The error of a numerical method may be much smaller for most instances than for the worst case. Also, two numerical methods may have the same maximal error although one of them usually is much better than the other. Such statements can be made precise by concepts from average case analysis. We give some examples where such an average case analysis seems to be more sensible than a worst case analysis.     

A generalization of the Lee distance and error correcting codes

In this contribution, we will define an l-dimensional Lee distance which is a generalization of the Lee distance defined only over a prime field, and we will construct 2-error correcting codes for this distance. Our l-dimensional Lee distance can be defined not only over a prime field but also over any finite field. The ordinary Lee distance is just the one-dimensional Lee distance. Also the Mannheim or modular distances introduced by Huber are special cases of our distance.     

Modular retractions of numerical semigroups

Let S be a numerical semigroup, let m be a nonzero element of S, and let a be a nonnegative integer. We denote ${\rm R}(S,a,m) = \{ s-as \bmod m \mid s \in S \}$ (where asmodm is the remainder of the division of as by m). In this paper we characterize the pairs (a,m) such that ${\rm R}(S,a,m)$ is a numerical semigroup. In this way, if we have a pair (a,m) with such characteristics, then we can reduce the problem of computing the genus of S=〈n ₁,…,n _p 〉 to computing the genus of a “smaller” numerical semigroup 〈n ₁?an ₁modm,…,n _p ?an _p modm〉. This reduction is also useful for estimating other important invariants of S such as the Frobenius number and the type.     

Modular translations of numerical semigroups

In this paper we introduce the concept of modular translation. With this tool, if we consider a certain numerical semigroup S, we build another one S′ whose principal invariants are given explicitly in terms of the invariants of S. Some results about irreducible numerical semigroups are also studied.     

Error and erasure correcting algorithms for rank codes

In this paper, transmitted signals are considered as square matrices of the Maximum rank distance (MRD) (n, k, d)-codes. A new composed decoding algorithm is proposed to correct simultaneously rank errors and rank erasures. If the rank of errors and erasures is not greater than the Singleton bound, then the algorithm gives always the correct decision. If it is not a case, then the algorithm gives still the correct solution in many cases but some times the unique solution may not exist.     

Linear covering codes and error-correcting codes for limited-magnitude errors

The concepts of a linear covering code and a covering set for the limited-magnitude-error channel are introduced. A number of covering-set constructions, as well as some bounds, are given. In particular, optimal constructions are given for some cases involving small-magnitude errors. A problem of Stein is partially solved for these cases. Optimal packing sets and the corresponding error-correcting codes are also considered for some small-magnitude errors.