Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA family of codes with locality containing optimal codes
97
0
0.0
(
0
)
Added by
Cicero Carvalho
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Locally recoverable codes were introduced by Gopalan et al. in 2012, and in the same year Prakash et al. introduced the concept of codes with locality, which are a type of locally recoverable codes. In this work we introduce a new family of codes with locality, which are subcodes of a certain family of evaluation codes. We determine the dimension of these codes, and also bounds for the minimum distance. We present the true values of the minimum distance in special cases, and also show that elements of this family are optimal codes, as defined by Prakash et al.
rate research
Read More
In this work we present a class of locally recoverable codes, i.e. codes where an erasure at a position $P$ of a codeword may be recovered from the knowledge of the entries in the positions of a recovery set $R_P$. The codes in the class that we define have availability, meaning that for each position $P$ there are several distinct recovery sets. Also, the entry at position $P$ may be recovered even in the presence of erasures in some of the positions of the recovery sets, and the number of supported erasures may vary among the various recovery sets.
Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[2^m+1, 2u-1, 2^m-2u+3]$ MDS codes for $1 leq u leq 2^{m-1}$, which are cyclic, reversible and BCH codes over $mathrm{GF}(2^m)$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, infinite families of $3$-designs from these quaternary codes are presented.
We study Algebraic Geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes. We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them.
Extending work of M. Zarzar, we evaluate the potential of Goppa-type evaluation codes constructed from linear systems on projective algebraic surfaces with small Picard number. Putting this condition on the Picard number provides some control over the numbers of irreducible components of curves on the surface and hence over the minimum distance of the codes. We find that such surfaces do not automatically produce good codes; the sectional genus of the surface also has a major influence. Using that additional invariant, we derive bounds on the minimum distance under the assumption that the hyperplane section class generates the Neron-Severi group. We also give several examples of codes from such surfaces with minimum distance better than the best known bounds in Grassls tables.
We propose a framework to study the effect of local recovery requirements of codeword symbols on the dimension of linear codes, based on a combinatorial proxy that we call emph{visible rank}. The locality constraints of a linear code are stipulated by a matrix $H$ of $star$s and $0$s (which we call a stencil), whose rows correspond to the local parity checks (with the $star$s indicating the support of the check). The visible rank of $H$ is the largest $r$ for which there is a $r times r$ submatrix in $H$ with a unique generalized diagonal of $star$s. The visible rank yields a field-independent combinatorial lower bound on the rank of $H$ and thus the co-dimension of the code. We prove a rank-nullity type theorem relating visible rank to the rank of an associated construct called emph{symmetric spanoid}, which was introduced by Dvir, Gopi, Gu, and Wigderson~cite{DGGW20}. Using this connection and a construction of appropriate stencils, we answer a question posed in cite{DGGW20} and demonstrate that symmetric spanoid rank cannot improve the currently best known $widetilde{O}(n^{(q-2)/(q-1)})$ upper bound on the dimension of $q$-query locally correctable codes (LCCs) of length $n$. We also study the $t$-Disjoint Repair Group Property ($t$-DRGP) of codes where each codeword symbol must belong to $t$ disjoint check equations. It is known that linear $2$-DRGP codes must have co-dimension $Omega(sqrt{n})$. We show that there are stencils corresponding to $2$-DRGP with visible rank as small as $O(log n)$. However, we show the second tensor of any $2$-DRGP stencil has visible rank $Omega(n)$, thus recovering the $Omega(sqrt{n})$ lower bound for $2$-DRGP. For $q$-LCC, however, the $k$th tensor power for $kle n^{o(1)}$ is unable to improve the $widetilde{O}(n^{(q-2)/(q-1)})$ upper bound on the dimension of $q$-LCCs by a polynomial factor.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
