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A block code is called a locally recoverable code (LRC for short) with $(r,\delta)$ -locality, if subject to any $\delta -1$ erasure failures, every symbol in the encoding can still be recovered by accessing at most $r$ other symbols. Recently, it was discovered by several authors that a $q$ -ary optimal $(r,\delta)$ -LRC, i.e., an LRC achieving the generalized 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 cases). In this paper, we further investigate constructions of optimal $(r,\delta)$ -LRCs along the line of using parity-check matrices. Inspired by classical Reed-Solomon codes and the work in Jin (2019), we equip parity-check matrices with the Vandermonde structure. It turns out that a parity-check matrix with the Vandermonde structure that produces an optimal LRC must obey certain disjoint property for subsets of $\mathbb {F}_{q}$. We manage to show the existence of these disjoint subsets. Thus, this yields an optimal $(r,\delta)$ -LRC with code length $\Omega \left({q^{\frac {\delta }{2}\left({1+\frac {1}{\lfloor \frac {d-1}{\delta }\rfloor -1}}\right)}}\right)\vphantom {\bigg)_{j}}$ , where $d$ is the minimum distance. In particular, to our surprise, for $\delta =2$ 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 Guruswami et al. (2018) for $d\geq 7$. In addition, for $d=6$ , we are able to remove the constraint required in Jin (2019) that $q$ is even. [ABSTRACT FROM AUTHOR]

We derive an outer bound of the rate region of the vector Gaussian \(L\) -terminal CEO problem by establishing a lower bound on each supporting hyperplane of the rate region. To this end, we prove a new extremal inequality by exploiting the connection between differential entropy and Fisher information as well as some fundamental estimation-theoretic inequalities. It is shown that the outer bound matches the Berger–Tung inner bound in the high-resolution regime. We then derive a lower bound on each supporting hyperplane of the rate region of the direct vector Gaussian \(L\) -terminal source coding problem by coupling it with the CEO problem through a limiting argument. The tightness of this lower bound in the high-resolution regime and the weak-dependence regime is also proved. [ABSTRACT FROM AUTHOR]

In this paper we classify all extremal and s-extremal binary self-dual codes of length 38. There are exactly 2744 extremal [38,19,8] self-dual codes, two s-extremal [38,19,6] codes, and 1730 s-extremal [38,19,8] codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of s-extremal [38,19,6] codes permits to achieve the classification of all s-extremal codes with d=6. [ABSTRACT FROM AUTHOR]

We explain how the classical notions of Fisher information of a random variable and Fisher information matrix of a random vector can be extended to a much broader setting. We also show that Stam's inequality for Fisher information and Shannon entropy, as well as the more generalized versions proved earlier by the authors, are all special cases of more general sharp inequalities satisfied by random vectors. The extremal random vectors, which we call generalized Gaussians, contain Gaussians as a limiting case but are noteworthy because they are heavy-tailed. [ABSTRACT FROM AUTHOR]

Let C be a binary extremal self-dual code of length 96. We prove that an automorphism of C of order 3 has 6 or no fixed points and an automorphism of order 5 has 6 fixed points. Moreover, if all automorphisms of order 3 are fixed point free then Aut(C) is solvable and its order divides 2^53 or 2^55 or Aut(C) is the alternating group A5 which is the only possible group of order 60. Furthermore, \vert Aut(C)\vert = 20 or 40 cannot occur. [ABSTRACT FROM AUTHOR]

We prove that the automorphism group of a binary self-dual doubly even [72, 36, 16] code has order 5, 7, 10, 14 or d where d divides 18 or 24, or it is A4 \times C3. [ABSTRACT FROM AUTHOR]

The capacity region of a compound multiple-antenna broadcast channel is characterized when the users exhibit a certain degradedness order. The channel under consideration has two users, each user has a finite set of possible realizations. The transmitter transmits two messages, one for each user, in such a manner that regardless of the actual realizations, both users will be able to decode their messages correctly. An alternative view of this channel is that of a broadcast channel with two common messages, each common message is intended to a different set of users. The degradedness order between the two sets of realizations/users is defined through an additional, fictitious, user whose channel is degraded with respect to all realizations/users from one set while all realizations/ users from the other set are degraded with respect to him. [ABSTRACT FROM AUTHOR]

In this correspondence, we present some results concerning a decomposition of binary self-dual codes possessing an automorphism of certain type. Two applications are given. The first one is the classification of all extremal doubly even self-dual codes of length 80 having an automorphism of order 39. The second one is the proof for uniqueness of the doubly even self-dual [80,40, 16] code having an automorphism of order 35. [ABSTRACT FROM AUTHOR]

The pth moment matrix is defined for a real random vector, generalizing the classical covariance matrix. Sharp inequalities relating the pth moment and Renyi entropy are established, generalizing the classical inequality relating the second moment and the Shannon entropy. The extremal distributions for these inequalities are completely characterized. [ABSTRACT FROM AUTHOR]

Focuses on a study which presented a method to determine the complete coset weight distributions of doubly even binary self-dual extremal codes. Information on the strategy and sequence of the tools; Details on a combinatorial formula; Explanation on obtaining the complete coset weight distribution of a binary self-dual extremal code.

Evaluates extremal self-dual codes with lengths 44, 50, 54 and 58 with weight enumerators for which extremal codes were previously not known to exist. Combinatorial optimization constructions of extremal self-dual codes; Results.

Investigates binary extremal self-dual codes. Update to length 64; Type II codes of lengths greater than 64; Codes with minimum weight 14.

Describes the weight enumerators for extremal singly-even self-dual codes. Extremal self-dual codes of length 36; Extremal self-dual codes of length 38.

One of the problems in coding theory is constructing self-dual codes whose weight enumerators are not yet known to exist. In this corre- spondence, we use the concept of neighbors and construct extremal binary self-dual [44, 22, 8] codes whose weight enumerators are not yet known to exist. [ABSTRACT FROM AUTHOR]