Weight Enumerators Research Articles

Some restrictions on the weight enumerators of near-extremal ternary self-dual codes and quaternary Hermitian self-dual codes

We give restrictions on the weight enumerators of ternary near-extremal self-dual codes of length divisible by 12 and quaternary near-extremal Hermitian self-dual codes of length divisible by 6. We consider the weight enumerators for which there is a ternary near-extremal self-dual code of length 12m for $$m =3,4,5,6$$ . Also we consider the weight enumerators for which there is a quaternary near-extremal Hermitian self-dual code of length 6m for $$m =4,5,6$$ .

On the complete weight distributions of quantum error-correcting codes

In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the MacWilliams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the MacWilliams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.

Weight enumerators of all irreducible cyclic codes of length n

Let [Formula: see text], where [Formula: see text] are distinct odd primes, be an integer and [Formula: see text] be a finite field of order [Formula: see text] with [Formula: see text]. We determine the weight enumerators of all irreducible cyclic codes of length [Formula: see text] over [Formula: see text] when multiplicative order of [Formula: see text] modulo [Formula: see text] is [Formula: see text]; [Formula: see text] and [Formula: see text]; [Formula: see text], where [Formula: see text].

Weight Enumerators and Cardinalities for Number-Theoretic Codes

The number-theoretic codes are a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts' codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo $r$.

On the binary codes of length 552 which admit the simple group Co3 as a transitive permutation group

In this this paper all binary codes of length 552 which admit the sporadic simple group as an imprimitive transitive permutation group are determined. Our aim is to understand the results also by using theoretical arguments and to discuss the combinatorial properties of the codes as well as their relation to some special properties of the Leech lattice group . For all codes (with two exceptions) we obtain the weight enumerators and in many interesting cases the classification of codewords under the action of the group of code automorphisms . The exceptional codes are both self-dual and have minimum weight 12.

Hamming polynomial of a demimatroid

Johnsen, Roksvold and Verdure have shown how fundamental invariants and concepts of linear coding theory, such as weight enumerators and higher weights, can be studied in the context of matroid theory. They have shown that Betti numbers, of a matroid and its elongations, associated to a linear code, determine both the weight enumerator and the higher weights of the matroid. In the same vein, Britz, Johnsen, Mayhew and Shiromoto, introduced demimatroids as a natural generalization of matroids. They have shown that demimatroids are the appropriate combinatorial objects for studying Wei’s duality. Continuing in this line of research, we study the Hamming polynomial of a demimatroid, which is a generalization of the extended weight enumerator of a matroid. It results that the Hamming polynomial is a specialization of the Tutte polynomial of the demimatroid, and actually is equivalent to it. In addition, guided by the work of Johnsen, Roksvold and Verdure for matroids, we prove that Betti numbers of, a demimatroid and its elongations, determine the Hamming polynomial. Our results may be applied to simplicial complexes since in a canonical way they can be viewed as demimatroids. Furthermore, following work of Brylawski and Gordon, we show how demimatroids may be generalized one step further, to Brylawski structures. All this largely extends notions such as deletion, contraction, duality, nullity, minors, weight enumerators, higher weights, Tutte polynomial and MacWilliams identity, to non-matroidal structures.

On the Construction of Self-Complementary Codes and their Application in the Problem of Information Hiding

Line codes are widely used to protect against errors in data transmission and storage systems, to ensure the stability of various cryptographic algorithms and protocols, to protect hidden information from errors in a stegocontainer. One of the classes of codes that find application in a number of the listed areas is the class of linear self-complementary codes over a binary field. Such codes contain a vector of all ones, and their weight enumerator is a symmetric polynomial. In applied problems, self-complementary [n, k]-codes are often required for a given length n and dimension k to have the maximum possible code distance d(k, n). For n < 13, the values of d(k, n) are already known. In this paper, for self-complementary codes of length n=13, 14, 15, the problem is to find lower bounds on d(k, n), as well as to find the values of d(k, n) themselves. The development of an efficient method for obtaining a lower estimate close to d(k, n) is an urgent task, since finding the values of d(k, n) in the general case is a difficult task. The paper proposes four methods for finding lower bounds: based on cyclic codes, based on residual codes, based on the (u-u+v)-construction, and based on the tensor product of codes. On the joint use of these methods for the considered lengths, it was possible to efficiently obtain lower bounds, either coinciding with the found values of d(k, n) or differing by one. The paper proposes a sequence of checks, which in some cases helps to prove the absence of a self-complementary [n, k]-code with code distance d. In the final part of the work, on the basis of self-complementary codes, a design for hiding information is proposed that is resistant to interference in the stegocontainer. The above calculations show the greater efficiency of the new design compared to the known designs.

Linear Programming Bounds for Approximate Quantum Error Correction Over Arbitrary Quantum Channels

While quantum weight enumerators establish some of the best upper bounds on the minimum distance of quantum error-correcting codes, these bounds are not optimized to quantify the performance of quantum codes under the effect of arbitrary quantum channels that describe bespoke noise models. Herein, for any Kraus decomposition of any given quantum channel, we introduce corresponding quantum weight enumerators that naturally generalize the Shor-Laflamme quantum weight enumerators. We establish an indirect linear relationship between these generalized quantum weight enumerators by introducing an auxiliary exact weight enumerator that completely quantifies the quantum code's projector, and is independent of the underlying noise process. By additionally working within the framework of approximate quantum error correction, we establish a general framework for constructing a linear program that is infeasible whenever approximate quantum error correcting codes with corresponding parameters do not exist. Our linear programming framework allows us to establish the non-existence of certain quantum codes that approximately correct amplitude damping errors, and obtain non-trivial upper bounds on the maximum dimension of a broad family of permutation-invariant quantum codes.

Narain CFTs and error-correcting codes on finite fields

We construct Narain CFTs from self-dual codes on the finite field Fp through even self-dual lattices for any prime p > 2. Using this correspondence, we can relate the spectral gap and the partition function of the CFT to the error correction capability and the extended enumerator polynomial of the code. In particular, we calculate specific spectral gaps of CFTs constructed from codes and compare them with the largest spectral gap among all Narain CFTs.

A Combinatorial Interpretation for the Shor-Laflamme Weight Enumerators of CWS Codes

We show that one of the Shor-Laflamme weight enumerators of a codeword stabilized quantum code may be interpreted as the distance enumerator of an associated classical code.

The extended coset leader weight enumerator of a twisted cubic code

The extended coset leader weight enumerator of the generalized Reed–Solomon [q+1,q-3,5]_q code is computed. In this computation methods in finite geometry, combinatorics and algebraic geometry are used. For this we need the classification of the points, lines and planes in the projective three space under projectivities that leave the twisted cubic invariant. A line in three space determines a rational function of degree at most three and vice versa. Furthermore, the double point scheme of a rational function is studied. The pencil of a true passant of the twisted cubic, not in an osculation plane gives a curve of genus one as double point scheme. With the Hasse–Weil bound on {mathbb F}_q-rational points we show that there is a 3-plane containing the passant.

Classical codes and chiral CFTs at higher genus

Higher genus modular invariance of two-dimensional conformal field theories (CFTs) is a largely unexplored area. In this paper, we derive explicit expressions for the higher genus partition functions of a specific class of CFTs: code CFTs, which are constructed using classical error-correcting codes. In this setting, the Sp(2g, ℤ) modular transformations of genus g Riemann surfaces can be recast as a simple set of linear maps acting on 2g polynomial variables, which comprise an object called the code enumerator polynomial. The CFT partition function is directly related to the enumerator polynomial, meaning that solutions of the linear constraints from modular invariance immediately give a set of seemingly consistent partition functions at a given genus. We then find that higher genus constraints, plus consistency under degeneration limits of the Riemann surface, greatly reduces the number of possible code CFTs. This work provides a step towards a full understanding of the constraints from higher genus modular invariance on 2d CFTs.

The weight distribution of hulls of binary codes from incidence matrices of complete graphs

We characterise the weights of all codewords of the hull of the binary code spanned by rows of an incidence matrix Bn of the complete graph Kn for n ≥ 4 and n even. Among others, we show that every codeword of the hull is the sum of an even number of rows of Bn and that its weight depends on the number of rows required to span it. We also determine the weight enumerator and the MacWilliams identity for the hull.

Weight enumeration in United States anti-bullying laws: associations with rates and risks of weight-based bullying among sexual and gender minority adolescents.

Weight-based bullying is among the most prevalent forms of peer harassment and carries significant health consequences-particularly among adolescents who identify as a sexual and/or gender minority (SGM). We examined how anti-bullying legislation that includes weight as a protected class (enumeration) contributes to the prevalence of weight-based bullying and its adverse health sequelae among SGM adolescents. We collected data on weight-based bullying and health risk (stress, unhealthy eating behaviors, self-rated health) from the LGBTQ National Teen Survey and linked these to state anti-bullying legislation obtained from the United States (US) Department of Health and Human Services. Weight-based bullying was less frequent, but associated with greater health risk for SGM adolescents in states with, versus without, weight-enumerated anti-bullying laws. Adding weight as a protected class in anti-bullying legislation may be an effective strategy for reducing weight-based bullying, but additional supports are needed to support adolescents who continue to experience weight-based bullying.

Another expression of the MacWilliams identities and its applications

&lt;p style='text-indent:20px;'&gt;Let &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathcal C $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; be a maximum distance separable (MDS) linear code over a finite field &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \Bbb F_q $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. In this paper, we present a new formula of its weight distribution, which can be seen as another expression of the MacWilliams identities. Moreover, using the new formula, we obtain the &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ b $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-symbol weight distribution of &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \mathcal C $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in the cases: &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ b = 2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ b = 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, respectively.&lt;/p&gt;

Infinite families of $ t $-designs and strongly regular graphs from punctured codes

&lt;p style='text-indent:20px;'&gt;The puncturing technique is sometimes efficient in constructing projective codes from original codes which are not projective. In this paper, several families of projective linear codes punctured from reducible cyclic codes, special linear codes or irreducible cyclic codes are investigated. The parameters and weight enumerators of the punctured codes and their duals are explicitly determined. Some of the codes are optimal and some of the codes are self-orthogonal which can be used to construct quantum codes. Several infinite families of combinatorial &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ 2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-designs and &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ 3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-designs including some families of Steiner systems are constructed from the punctured codes and their duals. Besides, infinite families of strongly regular graphs are also derived from some families of two-weight projective codes.&lt;/p&gt;

Self-orthogonal quasi-abelian codes are asymptotically good

Let F be the finite field with q=ps elements, where p is an odd prime and s is a positive integer. Suppose that gq−1(x) is the inverse function of gq(x)=1−hq(x), where hq(x) is the q-ary entropy. In this paper we construct a class of random self-orthogonal quasi-abelian codes of index 2p over the finite field F, characterize the cumulative weight enumerator of such random codes by means of a blend of representation theory and probabilistic arguments, and then prove that for any given δ∈(0,gq−1(1p)), the probability that the cumulative weight enumerator is at most δ converges to 0. As a consequence, the class of self-orthogonal quasi-abelian codes of index 2p is asymptotically good.

Self-orthogonal codes over a non-unital ring and combinatorial matrices

There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as \(E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle .\) We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over E, and Type IV codes, that is, quasi self-dual codes whose all codewords have even Hamming weight. All these codes can be represented as formally self-dual additive codes over \(\mathbb {F}_4.\) The classical invariant theory bound for the weight enumerators of this class of codes improves the known bound on the minimum distance of Type IV codes over E.

A formula on the weight distribution of linear codes with applications to AMDS codes

The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singleton's defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as extremal codes, Hermitian codes, MDS and NMDS codes. Moreover, we are able to present for the first time the weight distribution of AMDS codes. We also discuss the link between our results and the Pless equations.

New singly and doubly even binary [72,36,12] self-dual codes from M2(R)G - group matrix rings

In this work, we present a number of generator matrices of the form [I2n|τ2(v)], where I2n is the 2n×2n identity matrix, v is an element in the group matrix ring M2(R)G and where R is a finite commutative Frobenius ring and G is a finite group of order 18. We employ these generator matrices and search for binary [72,36,12] self-dual codes directly over the finite field F2. As a result, we find 134 Type I and 1 Type II codes of this length, with parameters in their weight enumerators that were not known in the literature before. We tabulate all of our findings.