42 research outputs found
List decoding group homomorphisms between supersolvable groups
We show that the set of homomorphisms between two supersolvable groups can be
locally list decoded up to the minimum distance of the code, extending the
results of Dinur et al who studied the case where the groups are abelian.
Moreover, when specialized to the abelian case, our proof is more streamlined
and gives a better constant in the exponent of the list size. The constant is
improved from about 3.5 million to 105.Comment: 11 page
List decoding Reed-Muller codes over small fields
The list decoding problem for a code asks for the maximal radius up to which
any ball of that radius contains only a constant number of codewords. The list
decoding radius is not well understood even for well studied codes, like
Reed-Solomon or Reed-Muller codes.
Fix a finite field . The Reed-Muller code
is defined by -variate degree-
polynomials over . In this work, we study the list decoding radius
of Reed-Muller codes over a constant prime field ,
constant degree and large . We show that the list decoding radius is
equal to the minimal distance of the code.
That is, if we denote by the normalized minimal distance of
, then the number of codewords in any ball of
radius is bounded by independent
of . This resolves a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008],
who among other results proved it in the special case of
; and extends the work of Gopalan [FOCS 2010] who
proved the conjecture in the case of .
We also analyse the number of codewords in balls of radius exceeding the
minimal distance of the code. For , we show that the number of
codewords of in a ball of radius is bounded by , where
is independent of . The dependence on is tight.
This extends the work of Kaufman-Lovett-Porat [IEEE Inf. Theory 2012] who
proved similar bounds over .
The proof relies on several new ingredients: an extension of the
Frieze-Kannan weak regularity to general function spaces, higher-order Fourier
analysis, and an extension of the Schwartz-Zippel lemma to compositions of
polynomials.Comment: fixed a bug in the proof of claim 5.6 (now lemma 5.5
Optimal Locally Repairable and Secure Codes for Distributed Storage Systems
This paper aims to go beyond resilience into the study of security and
local-repairability for distributed storage systems (DSS). Security and
local-repairability are both important as features of an efficient storage
system, and this paper aims to understand the trade-offs between resilience,
security, and local-repairability in these systems. In particular, this paper
first investigates security in the presence of colluding eavesdroppers, where
eavesdroppers are assumed to work together in decoding stored information.
Second, the paper focuses on coding schemes that enable optimal local repairs.
It further brings these two concepts together, to develop locally repairable
coding schemes for DSS that are secure against eavesdroppers.
The main results of this paper include: a. An improved bound on the secrecy
capacity for minimum storage regenerating codes, b. secure coding schemes that
achieve the bound for some special cases, c. a new bound on minimum distance
for locally repairable codes, d. code construction for locally repairable codes
that attain the minimum distance bound, and e. repair-bandwidth-efficient
locally repairable codes with and without security constraints.Comment: Submitted to IEEE Transactions on Information Theor
Tiny Codes for Guaranteeable Delay
Future 5G systems will need to support ultra-reliable low-latency
communications scenarios. From a latency-reliability viewpoint, it is
inefficient to rely on average utility-based system design. Therefore, we
introduce the notion of guaranteeable delay which is the average delay plus
three standard deviations of the mean. We investigate the trade-off between
guaranteeable delay and throughput for point-to-point wireless erasure links
with unreliable and delayed feedback, by bringing together signal flow
techniques to the area of coding. We use tiny codes, i.e. sliding window by
coding with just 2 packets, and design three variations of selective-repeat ARQ
protocols, by building on the baseline scheme, i.e. uncoded ARQ, developed by
Ausavapattanakun and Nosratinia: (i) Hybrid ARQ with soft combining at the
receiver; (ii) cumulative feedback-based ARQ without rate adaptation; and (iii)
Coded ARQ with rate adaptation based on the cumulative feedback. Contrasting
the performance of these protocols with uncoded ARQ, we demonstrate that HARQ
performs only slightly better, cumulative feedback-based ARQ does not provide
significant throughput while it has better average delay, and Coded ARQ can
provide gains up to about 40% in terms of throughput. Coded ARQ also provides
delay guarantees, and is robust to various challenges such as imperfect and
delayed feedback, burst erasures, and round-trip time fluctuations. This
feature may be preferable for meeting the strict end-to-end latency and
reliability requirements of future use cases of ultra-reliable low-latency
communications in 5G, such as mission-critical communications and industrial
control for critical control messaging.Comment: to appear in IEEE JSAC Special Issue on URLLC in Wireless Network
New Combinatorial Construction Techniques for Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes
This paper presents several new construction techniques for low-density
parity-check (LDPC) and systematic repeat-accumulate (RA) codes. Based on
specific classes of combinatorial designs, the improved code design focuses on
high-rate structured codes with constant column weights 3 and higher. The
proposed codes are efficiently encodable and exhibit good structural
properties. Experimental results on decoding performance with the sum-product
algorithm show that the novel codes offer substantial practical application
potential, for instance, in high-speed applications in magnetic recording and
optical communications channels.Comment: 10 pages; to appear in "IEEE Transactions on Communications
Hermitian self-dual quasi-abelian codes
Quasi-abelian codes constitute an important class of linear codes containing theoretically and practically interesting codes such as quasi-cyclic codes, abelian codes, and cyclic codes. In particular, the sub-class consisting of 1-generator quasi-abelian codes contains large families of good codes. Based on the well-known decomposition of quasi-abelian codes, the characterization and enumeration of Hermitian self-dual quasi-abelian codes are given. In the case of 1-generator quasi-abelian codes, we offer necessary and sufficient conditions for such codes to be Hermitian self-dual and give a formula for the number of these codes. In the case where the underlying groups are some -groups, the actual number of resulting Hermitian self-dual quasi-abelian codes are determined
Degree- Reverse Multiplication-Friendly Embeddings: Constructions and Applications
In the recent work of (Cheon & Lee, Eurocrypt\u2722), the concept of a degree- packing method was formally introduced, which captures the idea of embedding multiple elements of a smaller ring into a larger ring, so that element-wise multiplication in the former is somewhat compatible with the product in the latter.
Then, several optimal bounds and results are presented, and furthermore, the concept is generalized from one multiplication to degrees larger than two.
These packing methods encompass several constructions seen in the literature in contexts like secure multiparty computation and fully homomorphic encryption.
One such construction is the concept of reverse multiplication-friendly embeddings (RMFEs), which are essentially degree-2 packing methods.
In this work we generalize the notion of RMFEs to \emph{degree- RMFEs} which, in spite of being more algebraic than packing methods, turn out to be essentially equivalent.
Then, we present a general construction of degree- RMFEs by generalizing the ideas on algebraic geometry used to construct traditional degree- RMFEs which, by the aforementioned equivalence, leads to explicit constructions of packing methods.
Furthermore, our theory is given in an unified manner for general Galois rings, which include both rings of the form and fields like , which have been treated separately in prior works.
We present multiple concrete sets of parameters for degree- RMFEs (including ), which can be useful for future works.
Finally, we apply our RMFEs to the task of non-interactively generating high degree correlations for secure multiparty computation protocols.
This requires the use of Shamir secret sharing for a large number of parties, which is known to require large-degree Galois ring extensions.
Our RMFE enables the generation of such preprocessing data over small rings, without paying for the multiplicative overhead incurred by using Galois ring extensions of large degree.
For our application we also construct along the way, as a side contribution of potential independent interest, a pseudo-random secret-sharing solution for non-interactive generation of packed Shamir-sharings over Galois rings with structured secrets, inspired by the PRSS solutions from (Benhamouda et al, TCC 2021)
Graphical Structures for Design and Verification of Quantum Error Correction
We introduce a high-level graphical framework for designing and analysing
quantum error correcting codes, centred on what we term the coherent parity
check (CPC). The graphical formulation is based on the diagrammatic tools of
the zx-calculus of quantum observables. The resulting framework leads to a
construction for stabilizer codes that allows us to design and verify a broad
range of quantum codes based on classical ones, and that gives a means of
discovering large classes of codes using both analytical and numerical methods.
We focus in particular on the smaller codes that will be the first used by
near-term devices. We show how CSS codes form a subset of CPC codes and, more
generally, how to compute stabilizers for a CPC code. As an explicit example of
this framework, we give a method for turning almost any pair of classical
[n,k,3] codes into a [[2n - k + 2, k, 3]] CPC code. Further, we give a simple
technique for machine search which yields thousands of potential codes, and
demonstrate its operation for distance 3 and 5 codes. Finally, we use the
graphical tools to demonstrate how Clifford computation can be performed within
CPC codes. As our framework gives a new tool for constructing small- to
medium-sized codes with relatively high code rates, it provides a new source
for codes that could be suitable for emerging devices, while its zx-calculus
foundations enable natural integration of error correction with graphical
compiler toolchains. It also provides a powerful framework for reasoning about
all stabilizer quantum error correction codes of any size.Comment: Computer code associated with this paper may be found at
https://doi.org/10.15128/r1bn999672