相关论文: On $Z_{2^k}$-Dual Binary Codes
The main focus of this paper is the complete enumeration of self-dual abelian codes in non-principal ideal group algebras $\mathbb{F}_{2^k}[A\times \mathbb{Z}_2\times \mathbb{Z}_{2^s}]$ with respect to both the Euclidean and Hermitian inner…
In this work, we define a modification of a bordered construction for self-dual codes which utilises $\lambda$-circulant matrices. We provide the necessary conditions for the construction to produce self-dual codes over finite commutative…
Based on the generic construction of linear codes, we construct linear codes over the ring $\Bbb Z_4$ via posets of the disjoint union of two chains. We determine the Lee weight distributions of the quaternary codes. Moreover, we obtain…
Generalized Tanner graphs have been implicitly studied by a number of authors under the rubric of generalized parity-check matrices. This work considers the conditioning of binary hidden variables in such models in order to break all cycles…
A classic result of Delsarte connects the strength (as orthogonal array) of a linear code with the minimum weight of its dual: the former is one less than the latter. We show that Delsarte's observation extends to codes over arbitrary…
We give a new structural development of harmonic polynomials on Hamming space, and harmonic weight enumerators of binary linear codes, that parallels one approach to harmonic polynomials on Euclidean space and weighted theta functions of…
The generalized Hamming weights of a linear code have been extensively studied since Wei first use them to characterize the cryptography performance of a linear code over the wire-tap channel of type II. In this paper, we investigate the…
In this paper, we introduce some new polynomials associated to linear codes over $\mathbb{F}_{q}$. In particular, we introduce the notion of split complete Jacobi polynomials attached to multiple sets of coordinate places of a linear code…
We obtain all possible parameters of Plotkin-optimal two-Lee weight projective codes over $\mathbb{Z}_4,$ together with their weight distributions. We show the existence of codes with these parameters as well as their weight distributions…
Recently, minimal linear codes have been extensively studied due to their applications in secret sharing schemes, secure two-party computations, and so on. Constructing minimal linear codes violating the Ashikhmin-Barg condition and then…
Let $\mathbb{F}_{p^m}$ be the finite field of order $p^m$, where $p$ is an odd prime and $m$ is a positive integer. In this paper, we investigate a class of subfield codes of linear codes and obtain the weight distribution of…
Bachmann, Takeyama and Tasaka introduced a $\mathbb{Q}$-linear map $\phi$, which we call the symmetrization map in this paper, on the harmonic algebra $\mathfrak{H}^1$. They calculated $\phi(w)$ explicitly for an element $w$ in…
Binary field extensions are fundamental to many applications, such as multivariate public key cryptography, code-based cryptography, and error-correcting codes. Their implementation requires a foundation in number theory and algebraic…
The generalized Hamming weights (GHWs) of linear codes are fundamental parameters, the knowledge of which is of great interest in many applications. However, to determine the GHWs of linear codes is difficult in general. In this paper, we…
Binary codes are widely used to represent the data due to their small storage and efficient computation. However, there exists an ambiguity problem that lots of binary codes share the same Hamming distance to a query. To alleviate the…
Cyclic codes are an interesting family of linear codes since they have efficient decoding algorithms and contain optimal codes as subfamilies. Constructing infinite families of cyclic codes with good parameters is important in both theory…
In this paper, we have introduced the concepts of support distribution and the support enumerator as refinements of the classical weight distribution and weight enumerator respectively, capturing coordinate level activity in linear block…
We develop an algebraic theory of supports for $R$-linear codes of fixed length, where $R$ is a finite commutative unitary ring. A support naturally induces a notion of generalized weights and allows one to associate a monomial ideal to a…
Two-weight linear codes are linear codes in which any nonzero codeword can have only two possible distinct weights. Those in the Hamming metric have proven to be very interesting for their connections with authentication codes, association…
In most of the network coding problems with $k$ messages, the existence of binary network coding solution over $\mathbb{F}_2$ depends on the existence of adequate sets of $k$-dimensional binary vectors such that each set comprises of…