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Register a new userCharacterisation of the parameters of maximum weight spectrum codes according to their spread
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Added by
Alessio Meneghetti
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We introduce the concept of spread of a code, and we specialize it to the case of maximum weight spectrum (MWS) codes. We classify two newly-defined sub-families of MWS codes according to their weight distributions, and completely describe their fundamental parameters. We focus on one of these families, the strictly compact MWS codes, proving their optimality as MWS codes and linking them to known codes.
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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 Singletons 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.
We give a new, purely coding-theoretic proof of Kochs criterion on the tetrad systems of Type II codes of length 24 using the theory of harmonic weight enumerators. This approach is inspired by Venkovs approach to the classification of the root systems of Type II lattices in R^{24}, and gives a new instance of the analogy between lattices and codes.
Let $G$ be a directed graph such that the in-degree of any vertex $G$ is at least one. Let also ${mathcal{tau}}: V(G)rightarrow Bbb{N}$ be an assignment of thresholds to the vertices of $G$. A subset $M$ of vertices of $G$ is called a dynamic monopoly for $(G,tau)$ if the vertex set of $G$ can be partitioned into $D_0cup... cup D_t$ such that $D_0=M$ and for any $igeq 1$ and any $vin D_i$, the number of edges from $D_0cup... cup D_{i-1}$ to $v$ is at least $tau(v)$. One of the most applicable and widely studied threshold assignments in directed graphs is strict majority threshold assignment in which for any vertex $v$, $tau(v)=lceil (deg^{in}(v)+1)/2 rceil$, where $deg^{in}(v)$ stands for the in-degree of $v$. By a strict majority dynamic monopoly of a graph $G$ we mean any dynamic monopoly of $G$ with strict majority threshold assignment for the vertices of $G$. In this paper we first discuss some basic upper and lower bounds for the size of dynamic monopolies with general threshold assignments and then obtain some hardness complexity results concerning the smallest size of dynamic monopolies in directed graphs. Next we show that any directed graph on $n$ vertices and with positive minimum in-degree admits a strict majority dynamic monopoly with $n/2$ vertices. We show that this bound is achieved by a polynomial time algorithm. This upper bound improves greatly the best known result. The final note of the paper deals with the possibility of the improvement of the latter $n/2$ bound.
Ahlswede and Katona (1977) posed the following isodiametric problem in Hamming spaces: For every $n$ and $1le Mle2^{n}$, determine the minimum average Hamming distance of binary codes with length $n$ and size $M$. Fu, Wei, and Yeung (2001) used linear programming duality to derive a lower bound on the minimum average distance. However, their linear programming approach was not completely exploited. In this paper, we improve Fu-Wei-Yeungs bound by finding a better feasible solution to their dual program. For fixed $0<ale1/2$ and for $M=leftlceil a2^{n}rightrceil $, our feasible solution attains the asymptotically optimal value of Fu-Wei-Yeungs dual program as $ntoinfty$. Hence for $0<ale1/2$, all possible asymptotic bounds that can be derived by Fu-Wei-Yeungs linear program have been characterized. Furthermore, noting that the average distance of a code is closely related to weights of Fourier coefficients of a Boolean function, we also apply the linear programming technique to prove bounds on Fourier weights of a Boolean function of various degrees.
In this paper we consider the problem of determining the weight spectrum of q-ary codes C(3,m) associated with Grassmann varieties G(3,m). For m=6 this was done by Nogin. We derive a formula for the weight of a codeword of C(3,m), in terms of certain varieties associated with alternating trilinear forms on (F_q)^m. The classification of such forms under the action of the general linear group GL(m,F_q) is the other component that is required to calculate the spectrum of C(3,m). For m=7, we explicitly determine the varieties mentioned above. The classification problem for alternating 3-forms on (F_q)^7 was solved by Cohen and Helminck, which we then use to determine the spectrum of C(3,7).
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