11 research outputs found
Estimates for the range of binomiality in codes' spectra
We derive new estimates for the range of binomiality in a code’s spectra, where the distance distribution of a code is upperbounded by the corresponding normalized binomial distribution. The estimates
depend on the code’s dual distance
Correcting Charge-Constrained Errors in the Rank-Modulation Scheme
We investigate error-correcting codes for a the
rank-modulation scheme with an application to flash memory
devices. In this scheme, a set of n cells stores information in the
permutation induced by the different charge levels of the individual
cells. The resulting scheme eliminates the need for discrete
cell levels, overcomes overshoot errors when programming cells (a
serious problem that reduces the writing speed), and mitigates the
problem of asymmetric errors. In this paper, we study the properties
of error-correcting codes for charge-constrained errors in the
rank-modulation scheme. In this error model the number of errors
corresponds to the minimal number of adjacent transpositions required
to change a given stored permutation to another erroneous
one—a distance measure known as Kendall’s τ-distance.We show
bounds on the size of such codes, and use metric-embedding techniques
to give constructions which translate a wealth of knowledge
of codes in the Lee metric to codes over permutations in Kendall’s
τ-metric. Specifically, the one-error-correcting codes we construct
are at least half the ball-packing upper bound
Quantum Error-Control Codes
The article surveys quantum error control, focusing on quantum stabilizer
codes, stressing on the how to use classical codes to design good quantum
codes. It is to appear as a book chapter in "A Concise Encyclopedia of Coding
Theory," edited by C. Huffman, P. Sole and J-L Kim, to be published by CRC
Press
Quasi-cyclic Hermitian construction of binary quantum codes
In this paper, we propose a sufficient condition for a family of 2-generator
self-orthogonal quasi-cyclic codes with respect to Hermitian inner product.
Supported in the Hermitian construction, we show algebraic constructions of
good quantum codes. 30 new binary quantum codes with good parameters improving
the best-known lower bounds on minimum distance in Grassl's code tables
\cite{Grassl:codetables} are constructed
Algebraic Codes For Error Correction In Digital Communication Systems
Access to the full-text thesis is no longer available at the author's request, due to 3rd party copyright restrictions. Access removed on 29.11.2016 by CS (TIS).Metadata merged with duplicate record (http://hdl.handle.net/10026.1/899) on 20.12.2016 by CS (TIS).C. Shannon presented theoretical conditions under which communication was possible
error-free in the presence of noise. Subsequently the notion of using error
correcting codes to mitigate the effects of noise in digital transmission was introduced
by R. Hamming. Algebraic codes, codes described using powerful tools from
algebra took to the fore early on in the search for good error correcting codes. Many
classes of algebraic codes now exist and are known to have the best properties of
any known classes of codes. An error correcting code can be described by three of its
most important properties length, dimension and minimum distance. Given codes
with the same length and dimension, one with the largest minimum distance will
provide better error correction. As a result the research focuses on finding improved
codes with better minimum distances than any known codes.
Algebraic geometry codes are obtained from curves. They are a culmination of years
of research into algebraic codes and generalise most known algebraic codes. Additionally
they have exceptional distance properties as their lengths become arbitrarily
large. Algebraic geometry codes are studied in great detail with special attention
given to their construction and decoding. The practical performance of these codes
is evaluated and compared with previously known codes in different communication
channels. Furthermore many new codes that have better minimum distance
to the best known codes with the same length and dimension are presented from
a generalised construction of algebraic geometry codes. Goppa codes are also an
important class of algebraic codes. A construction of binary extended Goppa codes
is generalised to codes with nonbinary alphabets and as a result many new codes
are found. This construction is shown as an efficient way to extend another well
known class of algebraic codes, BCH codes. A generic method of shortening codes
whilst increasing the minimum distance is generalised. An analysis of this method
reveals a close relationship with methods of extending codes. Some new codes from
Goppa codes are found by exploiting this relationship. Finally an extension method
for BCH codes is presented and this method is shown be as good as a well known
method of extension in certain cases
Post-Quantum and Code-Based Cryptography—Some Prospective Research Directions
Cryptography has been used from time immemorial for preserving the confidentiality of data/information in storage or transit. Thus, cryptography research has also been evolving from the classical Caesar cipher to the modern cryptosystems, based on modular arithmetic to the contemporary cryptosystems based on quantum computing. The emergence of quantum computing poses a major threat to the modern cryptosystems based on modular arithmetic, whereby even the computationally hard problems which constitute the strength of the modular arithmetic ciphers could be solved in polynomial time. This threat triggered post-quantum cryptography research to design and develop post-quantum algorithms that can withstand quantum computing attacks. This paper provides an overview of the various research directions that have been explored in post-quantum cryptography and, specifically, the various code-based cryptography research dimensions that have been explored. Some potential research directions that are yet to be explored in code-based cryptography research from the perspective of codes is a key contribution of this paper
Near-capacity fixed-rate and rateless channel code constructions
Fixed-rate and rateless channel code constructions are designed for satisfying conflicting design tradeoffs, leading to codes that benefit from practical implementations, whilst offering a good bit error ratio (BER) and block error ratio (BLER) performance. More explicitly, two novel low-density parity-check code (LDPC) constructions are proposed; the first construction constitutes a family of quasi-cyclic protograph LDPC codes, which has a Vandermonde-like parity-check matrix (PCM). The second construction constitutes a specific class of protograph LDPC codes, which are termed as multilevel structured (MLS) LDPC codes. These codes possess a PCM construction that allows the coexistence of both pseudo-randomness as well as a structure requiring a reduced memory. More importantly, it is also demonstrated that these benefits accrue without any compromise in the attainable BER/BLER performance. We also present the novel concept of separating multiple users by means of user-specific channel codes, which is referred to as channel code division multiple access (CCDMA), and provide an example based on MLS LDPC codes. In particular, we circumvent the difficulty of having potentially high memory requirements, while ensuring that each user’s bits in the CCDMA system are equally protected. With regards to rateless channel coding, we propose a novel family of codes, which we refer to as reconfigurable rateless codes, that are capable of not only varying their code-rate but also to adaptively modify their encoding/decoding strategy according to the near-instantaneous channel conditions. We demonstrate that the proposed reconfigurable rateless codes are capable of shaping their own degree distribution according to the nearinstantaneous requirements imposed by the channel, but without any explicit channel knowledge at the transmitter. Additionally, a generalised transmit preprocessing aided closed-loop downlink multiple-input multiple-output (MIMO) system is presented, in which both the channel coding components as well as the linear transmit precoder exploit the knowledge of the channel state information (CSI). More explicitly, we embed a rateless code in a MIMO transmit preprocessing scheme, in order to attain near-capacity performance across a wide range of channel signal-to-ratios (SNRs), rather than only at a specific SNR. The performance of our scheme is further enhanced with the aid of a technique, referred to as pilot symbol assisted rateless (PSAR) coding, whereby a predetermined fraction of pilot bits is appropriately interspersed with the original information bits at the channel coding stage, instead of multiplexing pilots at the modulation stage, as in classic pilot symbol assisted modulation (PSAM). We subsequently demonstrate that the PSAR code-aided transmit preprocessing scheme succeeds in gleaning more information from the inserted pilots than the classic PSAM technique, because the pilot bits are not only useful for sounding the channel at the receiver but also beneficial for significantly reducing the computational complexity of the rateless channel decoder
Covering Radius 1985-1994
We survey important developments in the theory of covering radius during the period 1985-1994. We present lower bounds, constructions and upper bounds, the linear and nonlinear cases, density and asymptotic results, normality, specific classes of codes, covering radius and dual distance, tables, and open problems
Lengthening and the Gilbert-Varshamov bound
We use lengthening and an enhanced version of the Gilbert-Varshamov lower bound for linear codes to construct a large number of record-breaking codes. Our main theorem may be seen as a closure operation on databases. © 1997 IEEE