39,407 research outputs found
Signature Codes for a Noisy Adder Multiple Access Channel
In this work, we consider -ary signature codes of length and size
for a noisy adder multiple access channel. A signature code in this model has
the property that any subset of codewords can be uniquely reconstructed based
on any vector that is obtained from the sum (over integers) of these codewords.
We show that there exists an algorithm to construct a signature code of length
capable of correcting
errors at the channel output, where .
Furthermore, we present an explicit construction of signature codewords with
polynomial complexity being able to correct up to errors for a codeword length , where is a small non-negative
number. Moreover, we prove several non-existence results (converse bounds) for
-ary signature codes enabling error correction.Comment: 12 pages, 0 figures, submitted to 2022 IEEE Information Theory
Worksho
Bounds on the Sum Capacity of Synchronous Binary CDMA Channels
In this paper, we obtain a family of lower bounds for the sum capacity of
Code Division Multiple Access (CDMA) channels assuming binary inputs and binary
signature codes in the presence of additive noise with an arbitrary
distribution. The envelope of this family gives a relatively tight lower bound
in terms of the number of users, spreading gain and the noise distribution. The
derivation methods for the noiseless and the noisy channels are different but
when the noise variance goes to zero, the noisy channel bound approaches the
noiseless case. The behavior of the lower bound shows that for small noise
power, the number of users can be much more than the spreading gain without any
significant loss of information (overloaded CDMA). A conjectured upper bound is
also derived under the usual assumption that the users send out equally likely
binary bits in the presence of additive noise with an arbitrary distribution.
As the noise level increases, and/or, the ratio of the number of users and the
spreading gain increases, the conjectured upper bound approaches the lower
bound. We have also derived asymptotic limits of our bounds that can be
compared to a formula that Tanaka obtained using techniques from statistical
physics; his bound is close to that of our conjectured upper bound for large
scale systems.Comment: to be published in IEEE Transactions on Information Theor
Sign-Compute-Resolve for Tree Splitting Random Access
We present a framework for random access that is based on three elements:
physical-layer network coding (PLNC), signature codes and tree splitting. In
presence of a collision, physical-layer network coding enables the receiver to
decode, i.e. compute, the sum of the packets that were transmitted by the
individual users. For each user, the packet consists of the user's signature,
as well as the data that the user wants to communicate. As long as no more than
K users collide, their identities can be recovered from the sum of their
signatures. This framework for creating and transmitting packets can be used as
a fundamental building block in random access algorithms, since it helps to
deal efficiently with the uncertainty of the set of contending terminals. In
this paper we show how to apply the framework in conjunction with a
tree-splitting algorithm, which is required to deal with the case that more
than K users collide. We demonstrate that our approach achieves throughput that
tends to 1 rapidly as K increases. We also present results on net data-rate of
the system, showing the impact of the overheads of the constituent elements of
the proposed protocol. We compare the performance of our scheme with an upper
bound that is obtained under the assumption that the active users are a priori
known. Also, we consider an upper bound on the net data-rate for any PLNC based
strategy in which one linear equation per slot is decoded. We show that already
at modest packet lengths, the net data-rate of our scheme becomes close to the
second upper bound, i.e. the overhead of the contention resolution algorithm
and the signature codes vanishes.Comment: This is an extended version of arXiv:1409.6902. Accepted for
publication in the IEEE Transactions on Information Theor
Combinatorial Lower Bounds for 3-Query LDCs
A code is called a -query locally decodable code (LDC) if there is a
randomized decoding algorithm that, given an index and a received word
close to an encoding of a message , outputs by querying only at most
coordinates of . Understanding the tradeoffs between the dimension,
length and query complexity of LDCs is a fascinating and unresolved research
challenge. In particular, for -query binary LDCs of dimension and length
, the best known bounds are: .
In this work, we take a second look at binary -query LDCs. We investigate
a class of 3-uniform hypergraphs that are equivalent to strong binary 3-query
LDCs. We prove an upper bound on the number of edges in these hypergraphs,
reproducing the known lower bound of for the length of
strong -query LDCs. In contrast to previous work, our techniques are purely
combinatorial and do not rely on a direct reduction to -query LDCs, opening
up a potentially different approach to analyzing 3-query LDCs.Comment: 10 page
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