The structure of algebras admitting well-agreeing near weights

We characterize algebras admitting two well-agreeing near weights. These algebras can be used for constructing error-correcting codes of algebraic geometric type over two points, by using elementary methods only.     

Non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

Designs, Codes and Cryptography - In this paper, we construct a family of non-binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes. Moreover, we present a...     

On automorphism groups of certain Goppa codes

Goppa codes are linear codes arising from algebraic curves over finite fields. Sufficient conditions are given ensuring that all automorphisms of a Goppa code are inherited from the automorphism group of the curve. In some cases, these conditions are also necessary. The cases of curves with large automorphism groups, notably the Hermitian and the Deligne-Lusztig curves, are investigated in detail. This research was performed within the activity of GNSAGA of the Italian INDAM, with the financial support of the Italian Ministry MIUR, project “Strutture geometriche, combinatorica e loro applicazioni”, PRIN 2006–2007.     

On the duals of geometric Goppa codes from norm-trace curves

In this paper we study the dual codes of a wide family of evaluation codes on norm-trace curves. We explicitly find out their minimum distance and give a lower bound for the number of their minimum-weight codewords. A general geometric approach is performed and applied to study in particular the dual codes of one-point and two-point codes arising from norm-trace curves through Goppaʼs construction, providing in many cases their minimum distance and some bounds on the number of their minimum-weight codewords. The results are obtained by showing that the supports of the minimum-weight codewords of the studied codes obey some precise geometric laws as zero-dimensional subschemes of the projective plane. Finally, the dimension of some classical two-point Goppa codes on norm-trace curves is explicitely computed.     

The cardinality of functionally complete algebras on a three element set

It is known that there are continuum pairwise non-equivalent functionally complete algebras with the base set A if |A| ≥ 3. In this paper we prove the same statement for |A| = 3. Received December 20, 1996; accepted in final form July 3, 1997.     

On the Cyclicity of Goppa Codes,Parity-Check Subcodes of Goppa Codes,and Extended Goppa Codes

Classical Goppa codes are a special case of Alternant codes. First we prove that the parity-check subcodes of Goppa codes and the extended Goppa codes are both Alternant codes. Before this paper, all known cyclic Goppa codes were some particular BCH codes. Many families of Goppa codes with a cyclic extension have been found. All these cyclic codes are in fact Alternant codes associated to a cyclic Generalized Reed–Solomon code. In (1989, J. Combin. Theory Ser. A 51, 205–220) H. Stichtenoth determined all cyclic extended Goppa codes with this property. In a recent paper (T. P. Berger, 1999, in “Finite Fields: Theory, Applications and Algorithms (R. Mullin and G. Mullen, Eds.), pp. 143–154, Amer. Math. Soc., Providence), we used some semi-linear transformations on GRS codes to construct cyclic Alternant codes that are not associated to cyclic GRS codes. In this paper, we use these results to construct cyclic Goppa codes that are not BCH codes, new families of Goppa codes with a cyclic extension, and some families of non-cyclic Goppa codes with a cyclic parity-check subcode.     

On threefolds admitting a trigonal curve as abstract complete intersection

We study smooth projective varietiesX??^N of dimension 3, such that there are two very ample invertible sheaves ?, ? onX, and there exist two sections of theirs such that they intersect along a trigonal curveC. We give a classification of such threefolds under some hypotheses on the degree of the two embeddings given by ?, ?.     

Cayley-Bacharach and evaluation codes on complete intersections

Hansen (Appl. Algebra Eng. Comm. Comput. 14 (2003) 175) uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in P². In this paper, we generalize Hansen's results from P² to P^m; we also show that the hypotheses of Hansen (2003) may be weakened. The proof is succinct and follows by combining the Cayley-Bacharach Theorem and the bounds on evaluation codes obtained in Hansen (Zero-Dimensional Schemes (Ravello, 1992), de Gruyter, Berlin, 1994, pp. 205-211).     

Equality of geometric Goppa codes and equivalence of divisors

We give necessary and sufficient conditions for two geometric Goppa codes C_L(D,G) and C_L(D,H) to be the same. As an application we characterize self-dual geometric Goppa codes.     

On linear codes whose weights and length have a common divisor

In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least (r−1)q+(p−1)r points, where 1<r<q=ph, p prime. An immediate corollary of this theorem is that a linear code whose weights and length have a common divisor r<q and whose dual minimum distance is at least 3, has length at least (r−1)q+(p−1)r. The theorem, which is sharp in some cases, is a strong generalisation of an earlier result on the non-existence of maximal arcs in projective planes; the proof involves polynomials over finite fields, and is a streamlined and more transparent version of the earlier one.     

Lie algebras admitting a metacyclic frobenius group of automorphisms

Suppose that a Lie algebra L admits a finite Frobenius group of automorphisms FH with cyclic kernel F and complement H such that the characteristic of the ground field does not divide |H|. It is proved that if the subalgebra C _L (F) of fixed points of the kernel has finite dimension m and the subalgebra C _L (H) of fixed points of the complement is nilpotent of class c, then L has a nilpotent subalgebra of finite codimension bounded in terms of m, c, |H|, and |F| whose nilpotency class is bounded in terms of only |H| and c. Examples show that the condition of F being cyclic is essential.     

Algebraic-geometry codes, one-point codes, and evaluation codes

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed–Muller codes. Given an -algebra A, an order function on A and given a surjective -morphism of algebras , the ith evaluation code with respect to is defined as the code . In this work it is shown that under a certain hypothesis on the -algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G _i with associated divisors is the sequence of evaluation codes associated to some -algebra A, some order function and some surjective morphism with if and only if it is a sequence of one-point codes.      

On a complete decoding scheme for binary radical codes

This work was partially supported by the Fulbright commission, while the author was working at the Princeton University, Princeton NJ 08544, USA.     

Graded alphabets,circular codes,free Lie algebras and comma-free codes

We show how the use of graded alphabets allows one to provide simpler proofs of some results on free monoids and free Lie algebras. We first generalize to graded alphabets the characterization of the length distributions of circular codes. We also show that the existence of a circular code with a given distribution of degrees is equivalent to the existence of an embedding of Lie algebras. We finally give a generalization to graded alphabets of the famous result of Eastman on comma free codes of odd degree.     

On classifying monotone complete algebras of operators

We give a classification of “small” monotone complete C ^*-algebras by order properties. We construct a corresponding semigroup. This classification filters out von Neumann algebras; they are mapped to the zero of the classifying semigroup. We show that there are 2 ^c distinct equivalence classes (where c is the cardinality of the continuum). This remains true when the classification is restricted to special classes of monotone complete C ^*-algebras e.g. factors, injective factors, injective operator systems and commutative algebras which are subalgebras of ℓ^∞. Some examples and applications are given.