867 research outputs found
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
Locally Encodable and Decodable Codes for Distributed Storage Systems
We consider the locality of encoding and decoding operations in distributed
storage systems (DSS), and propose a new class of codes, called locally
encodable and decodable codes (LEDC), that provides a higher degree of
operational locality compared to currently known codes. For a given locality
structure, we derive an upper bound on the global distance and demonstrate the
existence of an optimal LEDC for sufficiently large field size. In addition, we
also construct two families of optimal LEDC for fields with size linear in code
length.Comment: 7 page
Computationally Relaxed Locally Decodable Codes, Revisited
We revisit computationally relaxed locally decodable codes (crLDCs) (Blocki
et al., Trans. Inf. Theory '21) and give two new constructions. Our first
construction is a Hamming crLDC that is conceptually simpler than prior
constructions, leveraging digital signature schemes and an appropriately chosen
Hamming code. Our second construction is an extension of our Hamming crLDC to
handle insertion-deletion (InsDel) errors, yielding an InsDel crLDC. This
extension crucially relies on the noisy binary search techniques of Block et
al. (FSTTCS '20) to handle InsDel errors. Both crLDC constructions have binary
codeword alphabets, are resilient to a constant fraction of Hamming and InsDel
errors, respectively, and under suitable parameter choices have
poly-logarithmic locality and encoding length linear in the message length and
polynomial in the security parameter. These parameters compare favorably to
prior constructions in the poly-logarithmic locality regime
Locally Decodable Codes with Randomized Encoding
We initiate a study of locally decodable codes with randomized encoding.
Standard locally decodable codes are error correcting codes with a
deterministic encoding function and a randomized decoding function, such that
any desired message bit can be recovered with good probability by querying only
a small number of positions in the corrupted codeword. This allows one to
recover any message bit very efficiently in sub-linear or even logarithmic
time. Besides this straightforward application, locally decodable codes have
also found many other applications such as private information retrieval,
secure multiparty computation, and average-case complexity.
However, despite extensive research, the tradeoff between the rate of the
code and the number of queries is somewhat disappointing. For example, the best
known constructions still need super-polynomially long codeword length even
with a logarithmic number of queries, and need a polynomial number of queries
to achieve a constant rate. In this paper, we show that by using a randomized
encoding, in several models we can achieve significantly better rate-query
tradeoff. In addition, our codes work for both the standard Hamming errors, and
the more general and harder edit errors.Comment: 23 page
- …