Weight Enumerators Research Articles

Byte weight enumerators of codes over 𝔽<SUB align="right">p and modular forms over a totally real field

In this paper, we obtain Jacobi forms over kp from byte weight enumerators of self-dual codes over Fp, where p is an odd prime, kp is the totally real field of the pth cyclotomic field and Fp is the finite field of order p. We also determine Siegel modular forms of genus g (g ≥ 1 is an integer) over kp by substituting certain theta series into byte weight enumerators in genus g of self-dual codes over Fp for all p ∈ P, where the set P consists of all those odd primes p for which the ring of algebraic integers of kp is a Euclidean domain. Further, we define some partial Epstein zeta functions and derive their functional equation using the Mellin transform of the theta series.

Invariants of MacWilliams identity: weight and ideal distribution in poset spaces

The weight distribution of a poset space is determined by the distribution of ideals in the poset. In this paper, we prove the opposite relation: the distribution of ideals is completely determined by the weight distribution of a poset space.

Invariants of MacWilliams identity: weight and ideal distribution in poset spaces

The weight distribution of a poset space is determined by the distribution of ideals in the poset. In this paper, we prove the opposite relation: the distribution of ideals is completely determined by the weight distribution of a poset space.

Self-dual codes with an automorphism of order 13

Using a method for constructing binary self-dual codes having an automorphism of odd prime order \begin{document}$p$\end{document} we classify, up to equivalence, all singly-even self-dual \begin{document}$[78,39,14]$\end{document} , \begin{document}$[80,40,14]$\end{document} , \begin{document}$[82,41,14],$\end{document} and \begin{document}$[84,42,14]$\end{document} codes as well as all doubly-even \begin{document}$[80,40,16]$\end{document} codes for \begin{document}$p=13$\end{document} . The results show that there are exactly 1592 inequivalent binary self-dual \begin{document}$[78,39,14]$\end{document} codes with an automorphism of type \begin{document}$13-(6,0)$\end{document} and we found 6 new values of the parameter in the weight function thus increasing more than twice the number of known values. As for binary \begin{document}$[80,40]$\end{document} self-dual codes with an automorphism of type \begin{document}$13-(6,2)$\end{document} there are 162696 singly-even self-dual codes with minimum distance 14 and 195 doubly-even such codes with minimum distance 16. We also construct many new codes of lengths 82 and 84 with minimum distance 14. Most of the constructed codes for all lengths have weight enumerators for which the existence was not known before.

On defining generalized rank weights

This paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over $L$, where $L$ is a finite extension of a field $K$. This is a generalization of the case where $K=\mathbb{F}_q $ and $L={\mathbb{F}_{{q^m}}}$ of Gabidulin codes to arbitrary characteristic. We show equivalence to previous definitions, in particular the ones by Kurihara-Matsumoto-Uyematsu [12,13], Oggier-Sboui [16] and Ducoat [6]. As an application of the notion of generalized rank weights, we discuss codes that are degenerate with respect to the rank metric.

A Construction of Linear Codes Over ${\mathbb {F}}_{2^t}$ From Boolean Functions

In this paper, we present a construction of linear codes over ${\mathbb {F}}_{2^{t}}$ from Boolean functions, which is a generalization of Ding’s method. Based on this construction, we give two classes of linear codes $\tilde { {\mathcal {C}}}_{f}$ and ${\mathcal {C}}_{f}$ over ${\mathbb {F}}_{2^{t}}$ from a Boolean function $f: {\mathbb {F}}_{q}\rightarrow {\mathbb {F}} _{2}$ , where $q=2^{n}$ and ${\mathbb {F}}_{2^{t}}$ is some subfield of ${\mathbb {F}}_{q}$ . The complete weight enumerator of $\tilde { {\mathcal {C}}}_{f}$ can be easily determined from the Walsh spectrum of $f$ , while the weight distribution of the code ${\mathcal {C}}_{f}$ can also be easily settled. Particularly, the number of nonzero weights of $\tilde { {\mathcal {C}}}_{f}$ and ${\mathcal {C}}_{f}$ is the same as the number of distinct Walsh values of $f$ . As applications of this construction, we show several series of linear codes over ${\mathbb {F}}_{2^{t}}$ with two or three weights by using bent, semibent, monomial and quadratic Boolean function $f$ .

Complete weight enumerators of a class of linear codes

Recently, linear codes constructed from defining sets have been studied extensively. They may have a few weights if the defining set is chosen properly. Let m and t be positive integers. For an odd prime p, let r=pm and Tr be the absolute trace function from Fr to Fp. In this paper, for b∈Fp∗, we define Db=(x1,…,xt)∈Frt:Tr(x1+⋯+xt)=b, and determine the complete weight enumerator of a class of p-ary linear codes given by CDb={c(a1,a2,…,at):a1,…,at∈Fr},where c(a1,a2,…,at)=(Tr(a1x12+⋯+atxt2))(x1,…,xt)∈Db.Then we get their weight enumerators explicitly, which will give us several linear codes with a few weights. As a generalization of Wang et al. (arXiv:1512.03866), this paper extends the result of Ahn et al. (2016).

Complete weight enumerators of some irreducible cyclic codes

In this paper, we investigate the complete weight enumerators of two classes of irreducible cyclic codes. We present the explicit complete weight enumerator of the irreducible cyclic codes. Furthermore, we obtain a class of optimal constant composition codes from irreducible cyclic codes.

On codes over 𝓡 k, m and constructions for new binary self-dual codes

Abstract In this work, we study codes over the ring 𝓡 k, m = 𝔽2[u, v]/ 〈uk , vm , uv − vu〉, which is a family of Frobenius, characteristic 2 extensions of the binary field. We introduce a distance and duality preserving Gray map from 𝓡 k, m to F 2 k m $\mathbb{F}_2^{km}$ together with a Lee weight. After proving the MacWilliams identities for codes over 𝓡 k, m for all the relevant weight enumerators, we construct many binary self-dual codes as the Gray images of self-dual codes over 𝓡 k, m . In addition to many extremal binary self-dual codes obtained in this way, including a new construction for the extended binary Golay code, we find 175 new Type I binary self-dual codes of parameters [72, 36, 12] and 105 new Type II binary self-dual codes of parameter [72, 36, 12].

MacWilliams type identities on the Lee and Euclidean weights for linear codes over [formula omitted

Motivated by the works of Shiromoto [3] and Shi et al. [4], we study the existence of the MacWilliams type identities with respect to the Lee and Euclidean weight enumerators for linear codes over the ring Zℓ. Necessary and sufficient conditions for the existence of the MacWilliams type identities with respect to the Lee and Euclidean weight enumerators for linear codes over Zℓ are given. Some examples about such MacWilliams type identities are also presented.

Intersection sets, three-character multisets and associated codes

In this article we construct new minimal intersection sets in ${\mathrm{AG}}(r,q^2)$ sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in ${\mathrm{PG}}(r,q^2)$ with $r$ even and we also compute their weight distribution.

Complete weight enumerators of two classes of linear codes

Recently, linear codes with a few weights have been constructed and extensively studied. In this paper, for an odd prime p, we determine the complete weight enumerator of two classes of p-ary linear codes constructed from defining set. Our results show that the codes have at most seven weights and may have applications in secret sharing schemes.

Complete weight enumerators of a class of three-weight linear codes

Linear codes have been an interesting subject of study for many years. In this paper, for an odd prime p, a class of three-weight linear codes over \(\mathbb {F}_{p}\) are constructed and the complete weight enumerators of these linear codes are also settled. These codes have applications in authentication codes, association schemes and data storage systems.

On modules with cyclic socle

The extension problem for linear codes over modules with respect to Hamming weight was already settled in [J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc. 136 (2008) 699–706; Foundations of linear codes defined over finite modules: The extension theorem and MacWilliams identities, in Codes Over Rings, Series on Coding Theory and Cryptology, Vol. 6 (World Scientific, Singapore, 2009), pp. 124–190]. A similar problem arises naturally with respect to symmetrized weight compositions (SWC). In 2009, Wood proved that Frobenius bimodules have the extension property (EP) for SWC. More generally, in [N. ElGarem, N. Megahed and J. A. Wood, The extension theorem with respect to symmetrized weight compositions, in 4th Int. Castle Meeting on Coding Theory and Applications (2014)], it is shown that having a cyclic socle is sufficient for satisfying the property, while the necessity remained an open question. Here, landing in midway, a partial converse is proved. For a (not small) class of finite module alphabets, the cyclic socle is shown necessary to satisfy the EP. The idea is bridging to the case of Hamming weight through a new weight function. Note: All rings are finite with unity, and all modules are finite too. This may be re-emphasized in some statements. The convention for left homomorphisms is that inputs are to the left.

A construction of several classes of two-weight and three-weight linear codes

Linear codes constructed from defining sets have been extensively studied and may have a few nonzero weights if the defining sets are well chosen. Let $${\mathbb {F}}_q$$Fq be a finite field with $$q=p^m$$q=pm elements, where p is a prime and m is a positive integer. Motivated by Ding and Ding's recent work (IEEE Trans Inf Theory 61(11):5835---5842, 2015), we construct p-ary linear codes $${\mathcal {C}}_D$$CD by $$\begin{aligned} {\mathcal {C}}_D=\{{\mathbf {c}}(a,b)=\big (\text {Tr}_m(ax+by)\big )_{(x,y)\in D}: a, b \in {\mathbb {F}}_q\}, \end{aligned}$$CD={c(a,b)=(Trm(ax+by))(x,y)źD:a,bźFq},where $$D \subset {\mathbb {F}}_q^2$$DźFq2 and $$\text {Tr}_m$$Trm is the trace function from $${\mathbb {F}}_q$$Fq onto $${\mathbb {F}}_p$$Fp. In this paper, we will employ exponential sums to investigate the weight enumerators of the linear codes $${\mathcal {C}}_D$$CD, where $$D=\{(x, y) \in {\mathbb {F}}_q^2 \setminus \{(0,0)\}: \text {Tr}_m(x^{N_1}+y^{N_2})=0\}$$D={(x,y)źFq2\{(0,0)}:Trm(xN1+yN2)=0} for two positive integers $$N_1$$N1 and $$N_2$$N2. Several classes of two-weight and three-weight linear codes and their explicit weight enumerators are presented if $$N_1, N_2 \in \{1, 2, p^{\frac{m}{2}}+1\}$$N1,N2ź{1,2,pm2+1}. By deleting some coordinates, more punctured two-weight and three-weight linear codes $${\mathcal {C}}_{\overline{D}}$$CD¯ which include some optimal codes are derived from $${\mathcal {C}}_D$$CD.

Complete weight enumerators of two classes of linear codes

In this paper, we give the complete weight enumerators of two classes of linear codes over the finite field Fp$\mathbb {F}_{p}$, where p is a prime. These linear codes are the torsion codes of MacDonald codes over the finite non-chain ring Fp+vFp$\mathbb {F}_{p}+v\mathbb {F}_{p}$, where v2 = v. We also employ these linear codes to construct systematic authentication codes with new parameters.

On the MacWilliams Identity for Classical and Quantum Convolutional Codes

The weight generating functions associated with convolutional codes (CCs) are based on state space realizations or the weight adjacency matrices (WAMs). The MacWilliams identity for CCs on the WAMs was first established by Gluesing- Luerssen and Schneider in the case of minimal encoders, and generalized by Forney. We consider this problem in the viewpoint of constraint codes and obtain a simple and direct proof of this MacWilliams identity in the case of minimal encoders. For our purpose, we choose a different representation for the exact weight generating function (EWGF) of a block code, by defining it as a linear combination of orthonormal vectors in Dirac bra-ket notation. This representation provides great flexibility so that general split weight generating functions and their MacWilliams identities can be easily obtained from the MacWilliams identity for EWGFs. As a result, we also obtain the MacWilliams identity for the input-parity weight adjacency matrices of a systematic convolutional code and its dual. Finally, paralleling the development of the classical case, we establish the MacWilliams identity for quantum convolutional codes.

A construction of codes with linearity from two linear codes

Let źźp$\mathbb {F}_{p}$ be a finite field, where p is a prime. This paper presents a new construction of a code C$\mathcal C$ over źźp2$\mathbb {F}_{p^{2}}$ from two linear codes over źźp$\mathbb {F}_{p}$, which has źźp$\mathbb {F}_{p}$-linearity and its minimum distance is equal to its minimum nonzero weight. Based on our construction, we shall determine the weight enumerators of some codes defined by quadratic bent functions and simplex codes. It will be shown that these codes have a few nonzero weights.

Complete weight enumerators of some linear codes from quadratic forms

Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, a construction of q-ary linear codes with few weights employing general quadratic forms over the finite field Fq${\mathbb {F}}_{q}$ is proposed, where q is an odd prime power. This generalizes some earlier constructions of p-ary linear codes from quadratic bent functions over the prime field Fp${\mathbb {F}}_{p}$, where p is an odd prime. The complete weight enumerators of the resultant q-ary linear codes are also determined.

On the linear codes over the ring Rp

Some results are generalized on linear codes over [Formula: see text] in [15] to the ring [Formula: see text], where [Formula: see text] is an odd prime number. The Gray images of cyclic and quasi-cyclic codes over [Formula: see text] are obtained. The parameters of quantum error correcting codes are obtained from negacyclic codes over [Formula: see text]. A nontrivial automorphism [Formula: see text] on the ring [Formula: see text] is determined. By using this, the skew cyclic, skew quasi-cyclic, skew constacyclic codes over [Formula: see text] are introduced. The number of distinct skew cyclic codes over [Formula: see text] is given. The Gray images of skew codes over [Formula: see text] are obtained. The quasi-constacyclic and skew quasi-constacyclic codes over [Formula: see text] are introduced. MacWilliams identities of linear codes over [Formula: see text] are given.