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Register a new userBounds for the multilevel construction
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Added by
Sascha Kurz
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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One of the main problems in random network coding is to compute good lower and upper bounds on the achievable cardinality of the so-called subspace codes in the projective space $mathcal{P}_q(n)$ for a given minimum distance. The determination of the exact maximum cardinality is a very tough discrete optimization problem involving a huge number of symmetries. Besides some explicit constructions for textit{good} subspace codes several of the most success full constructions involve the solution of discrete optimization subproblems itself, which mostly have not been not been solved systematically. Here we consider the multilevel a.k.a. Echelon--Ferrers construction and given lower and upper bounds for the achievable cardinalities. From a more general point of view, we solve maximum clique problems in weighted graphs, where the weights can be polynomials in the field size $q$.
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A basic problem for constant dimension codes is to determine the maximum possible size $A_q(n,d;k)$ of a set of $k$-dimensional subspaces in $mathbb{F}_q^n$, called codewords, such that the subspace distance satisfies $d_S(U,W):=2k-2dim(Ucap W)ge d$ for all pairs of different codewords $U$, $W$. Constant dimension codes have applications in e.g. random linear network coding, cryptography, and distributed storage. Bounds for $A_q(n,d;k)$ are the topic of many recent research papers. Providing a general framework we survey many of the latest constructions and show up the potential for further improvements. As examples we give improved constructions for the cases $A_q(10,4;5)$, $A_q(11,4;4)$, $A_q(12,6;6)$, and $A_q(15,4;4)$. We also derive general upper bounds for subcodes arising in those constructions.
Secure codes are widely-studied combinatorial structures which were introduced for traitor tracing in broadcast encryption. To determine the maximum size of such structures is the main research objective. In this paper, we investigate the lower bounds for secure codes and their related structures. First, we give some improved lower bounds for the rates of $2$-frameproof codes and $overline{2}$-separable codes for slightly large alphabet size. Then we improve the lower bounds for the rate of some related structures, i.e., strongly $2$-separable matrices and $2$-cancellative set families. Finally, we give a general method to derive new lower bounds for strongly $t$-separable matrices and $t$-cancellative set families for $tge 3.$
Recently, it was discovered by several authors that a $q$-ary optimal locally recoverable code, i.e., a locally recoverable code archiving the Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the classical $q$-ary MDS codes where it is conjectured that the code length is upper bounded by $q+1$ (or $q+2$ for some special case). This discovery inspired some recent studies on length of an optimal locally recoverable code. It was shown in cite{LXY} that a $q$-ary optimal locally recoverable code is unbounded for $d=3,4$. Soon after, it was proved that a $q$-ary optimal locally recoverable code with distance $d$ and locality $r$ can have length $Omega_{d,r}(q^{1 + 1/lfloor(d-3)/2rfloor})$. Recently, an explicit construction of $q$-ary optimal locally recoverable codes for distance $d=5,6$ was given in cite{J18} and cite{BCGLP}. In this paper, we further investigate construction optimal locally recoverable codes along the line of using parity-check matrices. Inspired by classical Reed-Solomon codes and cite{J18}, we equip parity-check matrices with the Vandermond structure. It is turns out that a parity-check matrix with the Vandermond structure produces an optimal locally recoverable code must obey certain disjoint property for subsets of $mathbb{F}_q$. To our surprise, this disjoint condition is equivalent to a well-studied problem in extremal graph theory. With the help of extremal graph theory, we succeed to improve all of the best known results in cite{GXY} for $dgeq 7$. In addition, for $d=6$, we are able to remove the constraint required in cite{J18} that $q$ is even.
Strong external difference family (SEDF) and its generalizations GSEDF, BGSEDF in a finite abelian group $G$ are combinatorial designs raised by Paterson and Stinson [7] in 2016 and have applications in communication theory to construct optimal strong algebraic manipulation detection codes. In this paper we firstly present some general constructions of these combinatorial designs by using difference sets and partial difference sets in $G$. Then, as applications of the general constructions, we construct series of SEDF, GSEDF and BGSEDF in finite fields by using cyclotomic classes.
In this letter, we propose a progressive rate-filling method as a framework to study agile construction of multilevel polar-coded modulation. We show that the bit indices within each component polar code can follow a fixed, precomputed ranking sequence, e.g., the Polar sequence in the 5G standard, while their allocated rates (i.e., the number of information bits of each component polar code) can be fast computed by exploiting the target sum-rate approximation and proper rate-filling methods. In particular, we develop two rate-filling strategies based on the capacity and the rate considering the finite block-length effect. The proposed construction methods can be performed independently of the actual channel condition with ${Oleft(mright)}$ ($m$ denotes the modulation order) complexity and robust to diverse modulation and coding schemes in the 5G standard, which is a desired feature for practical systems.
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