823 research outputs found
On the Locality of Codeword Symbols
Consider a linear [n,k,d]_q code C. We say that that i-th coordinate of C has
locality r, if the value at this coordinate can be recovered from accessing
some other r coordinates of C. Data storage applications require codes with
small redundancy, low locality for information coordinates, large distance, and
low locality for parity coordinates. In this paper we carry out an in-depth
study of the relations between these parameters.
We establish a tight bound for the redundancy n-k in terms of the message
length, the distance, and the locality of information coordinates. We refer to
codes attaining the bound as optimal. We prove some structure theorems about
optimal codes, which are particularly strong for small distances. This gives a
fairly complete picture of the tradeoffs between codewords length, worst-case
distance and locality of information symbols.
We then consider the locality of parity check symbols and erasure correction
beyond worst case distance for optimal codes. Using our structure theorem, we
obtain a tight bound for the locality of parity symbols possible in such codes
for a broad class of parameter settings. We prove that there is a tradeoff
between having good locality for parity checks and the ability to correct
erasures beyond the minimum distance
On Locally Decodable Index Codes
Index coding achieves bandwidth savings by jointly encoding the messages
demanded by all the clients in a broadcast channel. The encoding is performed
in such a way that each client can retrieve its demanded message from its side
information and the broadcast codeword. In general, in order to decode its
demanded message symbol, a receiver may have to observe the entire transmitted
codeword. Querying or downloading the codeword symbols might involve costs to a
client -- such as network utilization costs and storage requirements for the
queried symbols to perform decoding. In traditional index coding solutions,
this 'client aware' perspective is not considered during code design. As a
result, for these codes, the number of codeword symbols queried by a client per
decoded message symbol, which we refer to as 'locality', could be large. In
this paper, considering locality as a cost parameter, we view index coding as a
trade-off between the achievable broadcast rate (codeword length normalized by
the message length) and locality, where the objective is to minimize the
broadcast rate for a given value of locality and vice versa. We show that the
smallest possible locality for any index coding problem is 1, and that the
optimal index coding solution with locality 1 is the coding scheme based on
fractional coloring of the interference graph. We propose index coding schemes
with small locality by covering the side information graph using acyclic
subgraphs and subgraphs with small minrank. We also show how locality can be
accounted for in conventional partition multicast and cycle covering solutions
to index coding. Finally, applying these new techniques, we characterize the
locality-broadcast rate trade-off of the index coding problem whose side
information graph is the directed 3-cycle.Comment: 10 pages, 1 figur
Locality in Index Coding for Large Min-Rank
An index code is said to be locally decodable if each receiver can decode its
demand using its side information and by querying only a subset of the
transmitted codeword symbols instead of observing the entire codeword. Local
decodability can be a beneficial feature in some communication scenarios, such
as when the receivers can afford to listen to only a part of the transmissions
because of limited availability of power. The locality of an index code is the
ratio of the maximum number of codeword symbols queried by a receiver to the
message length. In this paper we analyze the optimum locality of linear codes
for the family of index coding problems whose min-rank is one less than the
number of receivers in the network. We first derive the optimal trade-off
between the index coding rate and locality with vector linear coding when the
side information graph is a directed cycle. We then provide the optimal
trade-off achieved by scalar linear coding for a larger family of problems,
viz., problems where the min-rank is only one less than the number of
receivers. While the arguments used for achievability are based on known coding
techniques, the converse arguments rely on new results on the structure of
locally decodable index codes.Comment: Keywords: index codes, locality, min-rank, directed cycle, side
information grap
Locally Decodable Index Codes
An index code for broadcast channel with receiver side information is locally
decodable if each receiver can decode its demand by observing only a subset of
the transmitted codeword symbols instead of the entire codeword. Local
decodability in index coding is known to reduce receiver complexity, improve
user privacy and decrease decoding error probability in wireless fading
channels. Conventional index coding solutions assume that the receivers observe
the entire codeword, and as a result, for these codes the number of codeword
symbols queried by a user per decoded message symbol, which we refer to as
locality, could be large. In this paper, we pose the index coding problem as
that of minimizing the broadcast rate for a given value of locality (or vice
versa) and designing codes that achieve the optimal trade-off between locality
and rate. We identify the optimal broadcast rate corresponding to the minimum
possible value of locality for all single unicast problems. We present new
structural properties of index codes which allow us to characterize the optimal
trade-off achieved by: vector linear codes when the side information graph is a
directed cycle; and scalar linear codes when the minrank of the side
information graph is one less than the order of the problem. We also identify
the optimal trade-off among all codes, including non-linear codes, when the
side information graph is a directed 3-cycle. Finally, we present techniques to
design locally decodable index codes for arbitrary single unicast problems and
arbitrary values of locality.Comment: Accepted for publication in the IEEE Transactions on Information
Theory. Parts of this manuscript were presented at IEEE ISIT 2018 and IEEE
ISIT 2019. This arXiv manuscript subsumes the contents of arXiv:1801.03895
and arXiv:1901.0590
On taking advantage of multiple requests in error correcting codes
In most notions of locality in error correcting codes -- notably locally
recoverable codes (LRCs) and locally decodable codes (LDCs) -- a decoder seeks
to learn a single symbol of a message while looking at only a few symbols of
the corresponding codeword. However, suppose that one wants to recover r > 1
symbols of the message. The two extremes are repeating the single-query
algorithm r times (this is the intuition behind LRCs with availability,
primitive multiset batch codes, and PIR codes) or simply running a global
decoding algorithm to recover the whole thing. In this paper, we investigate
what can happen in between these two extremes: at what value of r does
repetition stop being a good idea? In order to begin to study this question we
introduce robust batch codes, which seek to find r symbols of the message using
m queries to the codeword, in the presence of erasures. We focus on the case
where r = m, which can be seen as a generalization of the MDS property.
Surprisingly, we show that for this notion of locality, repetition is optimal
even up to very large values of
Index Codes with Minimum Locality for Three Receiver Unicast Problems
An index code for a broadcast channel with receiver side information is locally decodable if every receiver
can decode its demand using only a subset of the codeword
symbols transmitted by the sender instead of observing the entire
codeword. Local decodability in index coding improves the error
performance when used in wireless broadcast channels, reduces
the receiver complexity and improves privacy in index coding.
The locality of an index code is the ratio of the number of
codeword symbols used by each receiver to the number message
symbols demanded by the receiver. Prior work on locality in
index coding have considered only single unicast and singleuniprior problems, and the optimal trade-off between broadcast
rate and locality is known only for a few cases. In this paper we
identify the optimal broadcast rate (including among non-linear
codes) for all three receiver unicast problems when the locality
is equal to the minimum possible value, i.e., equal to one. The
index code that achieves this optimal rate is based on a clique
covering technique and is well known. The main contribution
of this paper is in providing tight converse results by relating
locality to broadcast rate, and showing that this known index
coding scheme is optimal when locality is equal to one. Towards
this we derive several structural properties of the side information
graphs of three receiver unicast problems, and combine them
with information theoretic arguments to arrive at a converse
Cooperative Local Repair in Distributed Storage
Erasure-correcting codes, that support local repair of codeword symbols, have
attracted substantial attention recently for their application in distributed
storage systems. This paper investigates a generalization of the usual locally
repairable codes. In particular, this paper studies a class of codes with the
following property: any small set of codeword symbols can be reconstructed
(repaired) from a small number of other symbols. This is referred to as
cooperative local repair. The main contribution of this paper is bounds on the
trade-off of the minimum distance and the dimension of such codes, as well as
explicit constructions of families of codes that enable cooperative local
repair. Some other results regarding cooperative local repair are also
presented, including an analysis for the well-known Hadamard/Simplex codes.Comment: Fixed some minor issues in Theorem 1, EURASIP Journal on Advances in
Signal Processing, December 201
A family of optimal locally recoverable codes
A code over a finite alphabet is called locally recoverable (LRC) if every
symbol in the encoding is a function of a small number (at most ) other
symbols. We present a family of LRC codes that attain the maximum possible
value of the distance for a given locality parameter and code cardinality. The
codewords are obtained as evaluations of specially constructed polynomials over
a finite field, and reduce to a Reed-Solomon code if the locality parameter
is set to be equal to the code dimension. The size of the code alphabet for
most parameters is only slightly greater than the code length. The recovery
procedure is performed by polynomial interpolation over points. We also
construct codes with several disjoint recovering sets for every symbol. This
construction enables the system to conduct several independent and simultaneous
recovery processes of a specific symbol by accessing different parts of the
codeword. This property enables high availability of frequently accessed data
("hot data").Comment: Minor changes. This is the final published version of the pape
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