Applicable Algebra in Engineering Communication and Computing最新文献

The label code of a lattice plays a key role in the characterization of the lattice. Every lattice (Lambda) can be described in terms of a label code L and an orthogonal sublattice (Lambda ') such that (Lambda /Lambda 'cong L). We identify the binomial ideal associated to an integer lattice and then establish a relation between the ideal quotient of the lattice and its label code. Furthermore, we present the Gröbner basis of the well-known root lattice (D_n). As an application of the relation (I_{Lambda }=I_{Lambda '}+I_{L}), where (I_{Lambda },I_{Lambda '}) and (I_L) denote binomial ideals associated to (Lambda ,~Lambda ') and L, respectively, a linear label code of (D_n) is obtained using its Gröbner basis.

In this paper, we study dihedral codes with 1-dimensional hulls and we determine precisely when dihedral codes over finite fields with 1-dimensional hulls exist. Moreover, we show that these codes come canonically in pairs. We also introduce 1-dimensional linear complementary pairs of dihedral codes and examine the properties of this class of codes. As an application, we obtain 1-dimensional linear complementary pair of dihedral codes, which are either optimal or near optimal.

We show that from every skew-type Hadamard matrix of order 4t one can obtain a series of skew-type Hadamard matrices of order (2^{i+2}t), i a positive integer, whose binary linear codes are doubly even self-dual binary codes of length (2^{i+2}t). It is known that a doubly even self-dual binary code yields an even unimodular lattice. Hence, this construction of skew-type Hadamard matrices gives us a series of even unimodular lattices of rank (2^{i+2}t), i a positive integer. Furthermore, we provide a construction of doubly even self-dual binary codes from conference graphs.

In this work we present a standard model for Galois rings based on the standard model of their residual fields, that is, a a sequence of Galois rings starting with ({mathbb Z}_{p^r}) that coves all the Galois rings with that characteristic ring and such that there is an algorithm producing each member of the sequence whose input is the size of the required ring.

In this paper, we study the set (mathscr {L}(m,r)) of all numerical semigroups with multiplicity m and ratio r. In particular, we present some algorithms that compute all the elements of ( mathscr {L}(m,r)) with a given genus or with a given Frobenius number.

In 1974, Dillon introduced two significant classes of bent functions, namely the Maiorana–McFarland class and the Partial Spread class. In this article, we studied a new subclass of biquadratic Maiorana–McFarland type bent functions and presented a lower bound on the third-order nonlinearity of this class. The resulting lower bounds are better than the ones from the earlier bounds of Carlet (for all biquadratic Boolean functions) and Garg et al. (for a different subclass).

We construct every finite-dimensional irreducible representation of the simple Lie algebra of type ({textsf {E}}_{7}) whose highest weight is a nonnegative integer multiple of the dominant minuscule weight associated with the type ({textsf {E}}_{7}) root system. As a consequence, we obtain constructions of each finite-dimensional irreducible representation of the simple Lie algebra of type ({textsf {E}}_{6}) whose highest weight is a nonnegative integer linear combination of the two dominant minuscule ({textsf {E}}_{6})-weights. Our constructions are explicit in the sense that, if the representing space is d-dimensional, then a weight basis is provided such that all entries of the (d times d) representing matrices of the Chevalley generators are obtained via explicit, non-recursive formulas. To effect this work, we introduce what we call ({textsf {E}}_{6})- and ({textsf {E}}_{7})-polyminuscule lattices that analogize certain lattices associated with the famous special linear Lie algebra representation constructions obtained by Gelfand and Tsetlin.

A combinatorial problem concerning the maximum size of the (Hamming) weight set of an ([n,k]_q) linear code was recently introduced. Codes attaining the established upper bound are the Maximum Weight Spectrum (MWS) codes. Those ([n,k]_q) codes with the same weight set as (mathbb {F}_q^n) are called Full Weight Spectrum (FWS) codes. FWS codes are necessarily “short”, whereas MWS codes are necessarily “long”. For fixed k, q the values of n for which an ([n,k]_q)-FWS code exists are completely determined, but the determination of the minimum length M(H, k, q) of an ([n,k]_q)-MWS code remains an open problem. The current work broadens discussion first to general coordinate-wise weight functions, and then specifically to the Lee weight and a Manhattan like weight. In the general case we provide bounds on n for which an FWS code exists, and bounds on n for which an MWS code exists. When specializing to the Lee or to the Manhattan setting we are able to completely determine the parameters of FWS codes. As with the Hamming case, we are able to provide an upper bound on (M({mathscr {L}},k,q)) (the minimum length of Lee MWS codes), and pose the determination of (M({mathscr {L}},k,q)) as an open problem. On the other hand, with respect to the Manhattan weight we completely determine the parameters of MWS codes.