In this paper we review existing hard-decision decoding algorithms for product codes along with different post-processing techniques used in conjunction with the iterative decoder for product codes. We improve the decoder by Reddy and Robinson and use it to create a new post-processing technique. The performance of this new post-processing technique is evaluated through simulations, and these suggest

Let \(K={\mathbb {Q}}(\zeta _8)\) be the complex multiplication field over \({\mathbb {Q}}\) of extension degree 4. We give an integral lattice construction on \({\mathbb {Q}}(\zeta _8)\) induced from binary codes. We define a theta series using these lattices and discuss its relation with the complete weight enumerator of a binary code. If C is a binary Type II code of length l, we find that the complete

We consider projective Hjelmslev geometries over finite chain rings of length 2 with residue field of order q. In these geometries we introduce and investigate the so-called homogeneous arcs defined as multisets of points with respect to a special weight function. These arcs are associated with linearly representable q-ary codes. We establish a relation between the parameters of a linearly representable

It is well-known that each left ideal in a matrix rings over a finite field is generated by an idempotent matrix. In this work we compute the number of left ideals in these rings, the number of different idempotents generating each left ideal, and give explicitly a set of idempotent generators of all left ideals of a given rank. We then apply these results to give examples of left group codes that

In this paper, we present a new perspective of single server private information retrieval (PIR) schemes by using the notion of linear error-correcting codes. Many of the known single server schemes are based on taking linear combinations between database elements and the query elements. Using the theory of linear codes, we develop a generic framework that formalizes all such PIR schemes. This generic

Constructing permutation polynomials is a hot topic in finite fields, and permutation polynomials have many applications in different areas. Recently, several classes of permutation trinomials with index \(q+1\) over \(\mathbf{F}_{q^2}\) were constructed. In this paper, we mainly construct permutation trinomials with index \(q+1\) over \(\mathbf{F}_{q^2}\). By using monomials of \(\mu _{(q+1)/2}\)

In this paper, we report some interesting results on permutations on \({\mathbb {Z}}_{n}\), the ring of integers modulo n, having full differential uniformity. By full differential uniformity of a permutation f on \({\mathbb {Z}}_{n}\), we mean that the cardinality of the set \(\{x\in {\mathbb {Z}}_{n}: f(x+a)-f(x)=b\}\) is exactly n for some \(a,b\in {\mathbb {Z}}_{n}\setminus \{0\}\). We give a sufficient

In this work, we study codes generated by elements that come from group matrix rings. We present a matrix construction which we use to generate codes in two different ambient spaces: the matrix ring \(M_k(R)\) and the ring R, where R is the commutative Frobenius ring. We show that codes over the ring \(M_k(R)\) are one sided ideals in the group matrix ring \(M_k(R)G\) and the corresponding codes over

In this paper, we introduce \(\omega\)-constacyclic Type II duadic codes over \({\mathbb {F}}_4\) of length \(n=3p^t\), where p is a prime number, \(p \equiv 1\) (mod 6), and give their main properties. We also obtain a splitting of n such that extended \(\omega\)-constacyclic duadic codes are Hermitian self-dual. Moreover, we provide some examples by extended \(\omega\)-constacyclic duadic codes.

Let \((S,\le )\) be a strictly totally ordered monoid and R an \((S,\omega )\)-weakly rigid ring, where \(\omega :S\rightarrow End(R)\) is a monoid homomorphism. In this paper, we study the weakly p.q.-Bear property of the skew generalized power series ring \(R[[S,\omega ]]\). As a consequence, the weakly p.q.-Baer property of the skew power series ring \(R[[x;\alpha ]]\) and the skew Laurent power

Let \(\mathcal S\) be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of \(\mathcal S\) by means

It is well known that any finite commutative ring is isomorphic to a direct product of local rings via the Chinese remainder theorem. Hence, there is a great significance to the study of character sums over local rings. Character sums over finite rings have applications that are analogous to the applications of character sums over finite fields. In particular, character sums over local rings have many

For a Quillen exact category \({\mathcal {C}}\) endowed with two exact structures \({\mathcal {D}}\) and \({\mathcal {E}}\) such that \({\mathcal {E}}\subseteq {\mathcal {D}}\), an object X of \({\mathcal {C}}\) is called \({\mathcal {E}}\)-divisible (respectively \({\mathcal {E}}\)-flat) if every short exact sequence from \({\mathcal {D}}\) starting (respectively ending) with X belongs to \({\mathcal

In this note we investigate minimal linear codes arising from Hermitian varieties and quadrics. We study their parameters and formulate some open problems about their weight distribution.

Let \(\theta \) be an automorphism on a finite field \(\mathbb {F}_q.\) In this paper, we give a way to construct and enumerate Euclidean self-dual \(\theta \)-cyclic codes of length n over \(\mathbb {F}_q\) when n is even and \(\gcd (n,|\theta |)=1.\) The restriction \(\gcd (n,|\theta |)=1\) implies that the \(\theta \)-cyclic codes are in fact cyclic codes and \(q=2^m,\) for some integer \(m\ge 1

Linear codes with complementary duals, shortly named LCD codes, are linear codes whose intersection with their duals is trivial. In this paper, we outline a construction for LCD codes over finite fields from the adjacency matrices of two-class association schemes. These schemes consist of either strongly regular graphs (SRGs) or doubly regular tournaments (DRTs). Under certain conditions, the method

Structural parameter identifiability is a property of a differential model with parameters that allows for the parameters to be determined from the model equations in the absence of noise. One of the standard approaches to assessing this problem is via input–output equations and, in particular, characteristic sets of differential ideals. The precise relation between identifiability and input–output

The object of this article is associate to automorphism-invariant modules that are invariant under any automorphism of their injective hulls with cyclic modules and cyclic modules have cyclic automorphism-invariant hulls. The study of the first sequence allows us to characterize rings whose cyclic right modules are automorphism-invariant and to show that if R is a right Köthe ring, then R is an Artinian

In this paper we generalize the construction of binary self-orthogonal codes obtained from weakly self-orthogonal designs described in Tonchev (J Combinat Theory Ser A 52:197-205, 1989) in order to obtain self-orthogonal codes over an arbitrary field. We extend construction self-orthogonal codes from orbit matrices of self-orthogonal designs and weakly self-orthogonal 1-designs such that block size

Let \({\mathcal{R}}\) be the finite chain ring \({\mathcal{R}}={\mathbb{F}}_{p^{m}}+ u{\mathbb{F}}_{p^{m}}(u^{2} = 0)\), where p is an odd prime number and m is a positive integer. For \(\eta \in {\mathbb{F}}_{p^{m}}^{*}\), the Hamming distances of all \(\eta\)-constacyclic codes of length \(2p^{s}\) over \({\mathcal{R}}\) had already been studied in Dinh et al. (in AAECC, 2020. https://doi.org/10

Recall that a ring satisfies the 2-sum property if each of its elements is a sum of two units. Here a ring R is said to satisfy the binary 2-sum property if, for any a, b in R, there exists a unit u of R such that both \(a-u\) and \(b-u\) are units. A well-known result, due to Goldsmith, Pabst and Scot, states that a semilocal ring satisfies the 2-sum property iff it has no image isomorphic to \(\mathbb

We adapt tag-variables and Buchberger reduction in order, given two elements, \(pg\in R\) into an effective ring R, to express g as the evaluation of a polynomial \(f(X)\in R[X]\) at p, \(g=f(p)\). As a by-product, we present also an attack to a couple of Cryptographical protocols.

There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as $$\begin{aligned} I=\langle a,b \mid 2a=2b=0,\, a^{2}=b,\, \,ab=0 \rangle . \end{aligned}$$ We give a natural map between linear codes over I and additive codes over \({\mathbb{F}}_{4},\) that allows for efficient computations. We study the algebraic structure of linear codes over this

Let R be a commutative semiring with \(1\ne 0\), M an R-semimodule, and \(n \ge 1\) a positive integer. A proper subsemimodule P of M is called n-absorbing, if whenever \(r_1,\ldots ,r_n \in R\) and \(x \in M\) together with \(r_1 \ldots r_nx \in P\), imply \(r_1 \ldots r_n \in (P : M)\) or there exists \(1 \le i \le n\) such that \(r_1 \ldots \widehat{r_i} \ldots r_nx \in P\). In this paper, we study

As a subclass of linear codes, cyclic codes have efficient encoding and decoding algorithms, so they are widely used in many areas such as consumer electronics, data storage systems and communication systems. In this paper, we give a general construction of optimal p-ary cyclic codes which leads to three explicit constructions. In addition, another class of p-ary optimal cyclic codes are presented

The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in common. Lastly

Let \(\gamma = 4z-1\) be an unit of Type \((*^{-})\) of the Galois ring \({{\,\mathrm{GR}\,}}(2^a, m)\). The \(\gamma\)-constacyclic codes of length \(2^s\) over the Galois ring \({{\,\mathrm{GR}\,}}(2^a, m)\) are precisely the ideals \(\langle (x +1)^i \rangle\), \(0 \le i \le 2^sa\) of the chain ring \(\mathfrak {R}(a,m, \gamma ) = \dfrac{{{\,\mathrm{GR}\,}}(2^a,m)[x]}{\langle {x^{2^s}} - \gamma

Let \(m=8k\) and \(\alpha\) be a primitive element of the finite field \({{\mathbb {G}}{\mathbb {F}}}(2^m)\), where \(k\ge 2\) is an integer. In this paper, a class of binary cyclic codes \({{\mathcal {C}}}_{(u,v)}\) of length \(2^m-1\) with two nonzeros \(\alpha ^{-u}\) and \(\alpha ^{-v}\) is studied, where \((u,v)=(1,(2^{m}-1)/17)\). It turns out that \({{\mathcal {C}}}_{(u,v)}\) has parameters

In this paper, we give the exact number of \({{\mathbb {Z}}}_{2}{{\mathbb {Z}}}_{4}\)-additive cyclic codes of length \(n=r+s,\) for any positive integer r and any positive odd integer s. We will provide a formula for the the number of separable \({{\mathbb {Z}}_{2}{{\mathbb {Z}}_{4}}}\)-additive cyclic codes of length n and then a formula for the number of non-separable \({{\mathbb {Z}} _{2}{{\mathbb

In this paper, we continue to study and investigate the homological objects related to s-pure and neat exact sequences of modules and module homomorphisms. A right module A is called max-flat if \({\text {Tor}}_{1}^{R}(A, R/I)= 0\) for any maximal left ideal I of R. A right module B is said to be max-cotorsion if \({\text {Ext}}^{1}_{R}(A,B)=0\) for any max-flat right module A. We characterize some

In this paper we investigate some dual algebraic-geometric codes associated with the Giulietti–Korchmáros maximal curve. We compute the minimum distance and the minimum weight codewords of such codes and we investigate the generalized Hamming weights of such codes.

Let R be a ring such that \(R=R_1+R_2\), where \(R_1\) is a PI subring of R and \(R_2\) is an additive subgroup of R which satisfies a polynomial identity. We prove that if for some integer \(n\ge 1\) either \((R_1R_2)^n \subseteq R_1\) or \((R_2R_1)^n \subseteq R_1\), then R is a PI ring.

For \(m\ge 4\) even, the duals of p-ary codes, for any prime p, from adjacency matrices for the m-ary 2-cube \(Q^m_2\) are shown to have subcodes with parameters \([m^2,2m-2,m]\) for which minimal PD-sets of size \(\frac{m}{2}\) are constructed, hence attaining the full error-correction capabilities of the code, and, as such, the most efficient sets for full permutation decoding.

The study of topological invariants of finite topological spaces is relevant because they can be used as models of a wide class of topological spaces, including regular CW-complexes. In this work, we present a new module for the Kenzo system that allows the computation of homology groups with generators of finite topological spaces in different situations. Our algorithms combine new constructive versions

In this paper, we prove that flag-transitive automorphism groups of 2-designs with \(10^3\ge \lambda \ge (r,\lambda )^2\) are point-primitive of affine or almost simple type.

In this paper we derive formulas for the scalar multiplication by n map, denoted [n], on the Hessian model of elliptic curve. This enables to characterize n-torsion points on this curve. The computation involves three families of polynomials \(P_n\), \(Q_n\) and \(V_n\) and we show some properties on the coefficients and degrees of these polynomials. We also show some functional equations satisfied

Let \(\gamma\) be a generator of a cyclic group G of order n. The least index of a self-mapping f of G is the index of the largest subgroup U of G such that \(f(x)x^{-r}\) is constant on each coset of U for some positive integer r. We determine the index of the univariate Diffie–Hellman mapping \(d(\gamma ^a)=\gamma ^{a^2}\), \(a=0,1,\ldots ,n-1\), and show that any mapping of small index coincides

We define a self-dual code over a finite abelian group in terms of an arbitrary duality on the ambient space. We determine when additive self-dual codes exist over abelian groups for any duality and describe various constructions for these codes. We prove that there must exist self-dual codes under any duality for codes over a finite abelian group \({\mathbb {Z}}_{p^e}\). They exist for all lengths

Let \({\mathbb {F}}_q\) be a finite field with q elements and G be a finite abelian group. In this work we gave conditions to ensure that a code in \({\mathbb {F}}_qG\) is a one-weight code in the case when G is a cyclic group with n elements, such that \({\text {gcd}}(n,q) = 1\), and also when G is an abelian group.

A formal weight enumerator is a homogeneous polynomial in two variables which behaves like the Hamming weight enumerator of a self-dual linear code except that the coefficients are not necessarily nonnegative integers. The notion of formal weight enumerator was first introduced by Ozeki in connection with modular forms, and a systematic investigation of formal weight enumerators has been conducted

We propose a general greedy algorithm for binary de Bruijn sequences, called Generalized Prefer-Opposite Algorithm, and its modifications. By identifying specific feedback functions and initial states, we demonstrate that most previously-known greedy algorithms that generate binary de Bruijn sequences are particular cases of our algorithm.

Low-density parity-check (LDPC) codes have been the subject of much interest due to the fact that they can perform near the Shannon limit. In this paper we present a construction of LDPC codes from cubic symmetric graphs. The codes constructed are (3, 3)-regular and the vast majority of the corresponding Tanner graphs have girth greater than four. We analyse properties of the codes obtained and present

The extension problem underlies notions of code equivalence. Two approaches to the extension problem are described. One is a matrix approach that reduces the general problem for weights to one for symmetrized weight compositions. The other is a monoid algebra approach that reframes the extension problem in terms of modules over the monoid algebra determined by the multiplicative monoid of a finite

Traditional global stability measure for sequences is hard to determine because of large search space. We propose the k-error linear complexity with a zone restriction for measuring the local stability of sequences. For several classes of sequences, we demonstrate that the k-error linear complexity is identical to the k-error linear complexity within a zone, while the length of a zone is much smaller

Differential Convolutional Codes with designed Hamming distance are defined, and an algebraic decoding algorithm, inspired by Peterson–Gorenstein–Zierler’s algorithm, is designed for them.

We construct an infinite family of commutative rings \({R_{q,\varDelta }}\) and we study codes over these rings as well as the structure of the rings. We define a canonical Gray map from \({R_{q,\varDelta }}\) to vectors over the residue finite field of q elements and use it to relate codes over \({R_{q,\varDelta }}\) to codes over the finite field \({\mathbb {F}}_q\). Finally, we determine the parameters

Quasi-symmetric (36, 16, 12) designs with intersection numbers \(x=6\), \(y=8\) that cannot be embedded in symmetric (64, 28, 12) designs as residuals are constructed.

The support poset of a monomial ideal \(I\subseteq {{\mathbf {k}}}[x_1,\ldots ,x_n]\) encodes the relation between the variables \(x_1,\ldots ,x_n\) and the minimal monomial generators of I. It is known that not every poset is realizable as the support poset of some monomial ideal. We describe some posets P for which we can explicitly find at least one monomial ideal \(I_P\) such that P is the support

InfoMod is a software devoted to the modular group, \(\mathrm {PSL}_2 (\mathbf {Z})\). It consists of algorithms that deal with the classical correspondences among geodesics on the modular surface, elements of the modular group and binary quadratic forms. In addition, the software implements the recently discovered representation of Gauss’ indefinite binary quadratic forms and their classes in terms

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, by investigating the solutions of certain equations over finite fields, we make progress towards three conjectures about optimal ternary cyclic codes which were proposed by Ding and Helleseth

The q-analog of Kostant’s weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant’s partition function. This formula, when evaluated at \(q=1\), gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra \(\mathfrak {sl}_4(\mathbb {C})\)

For a prime p and positive integers m, n, let \({{\mathbb {F}}}_q\) be a finite field with \(q=p^m\) elements and \({{\mathbb {F}}}_{q^n}\) be an extension of \({{\mathbb {F}}}_q.\) Let h(x) be a polynomial over \({{\mathbb {F}}}_{q^n}\) satisfying the following conditions: (i) \({\mathrm{Tr}}_m^{nm}(x)\circ h(x)=\tau (x)\circ {\mathrm{Tr}}_m^{nm}(x)\); (ii) For any \(s \in {{\mathbb {F}}}_{q}\), h(x)

Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over \(\mathbb {C}\), this important data coincides with the topological class. In this paper, we characterise a family of singularities, containing irreducible ones, whose equisingularity type can be computed in an expected quasi-linear time with respect to the discriminant valuation of

In this paper we extend the theory of involutive divisions to the case of monomials with coefficients over effective rings. Moreover, as regards involutive bases, we study the computation of weak involutive bases and sketch a conjecture on strong involutive bases.

Symplectic self-orthogonal codes over finite fields are an important class of linear codes in coding theory, which can be used to construct quantum codes. In this paper, characterizations of symplectic self-orthogonal codes over finite fields \(F_{q}\) are given. A necessary and sufficient condition for determining symplectic self-orthogonal codes is obtained. Several classes of symplectic self-orthogonal

Let q be a prime power.

An associative memory is a special type of artificial neural network that has the purpose of store input patterns with their corresponding output patterns and efficiently recall a pattern from a noise-distorted version. Presented in this article is a new framework for constructing a binary associative memory model based on two new autoinverse operations called extended XOR/XNOR; these new operations

We present two algorithms determining all the complete and simplicial fans admitting a fixed non-degenerate set of vectors V as generators of their 1-skeleton. The interplay of the two algorithms allows us to discerning if the associated toric varieties admit a projective embedding, in principle for any values of dimension and Picard number. The first algorithm is slower than the second one, but it

Rigid graph theory is an active area with many open problems, especially regarding embeddings in \({\mathbb R}^d\) or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system’s complex solutions

The F4 algorithm re-imagines Buchberger’s algorithm as the row reduction of a Macaulay matrix: each row corresponds to a polynomial; each reduction of one row by another corresponds to one step of reducing an S-polynomial; and any row that completes reduction with a new pivot position corresponds to a new element of the basis. On the other hand, each column corresponds to a term, so that while it is