76 research outputs found

### Systems of MDS codes from units and idempotents

Algebraic systems are constructed from which series of maximum distance
separable (mds) codes are derived. The methods use unit and idempotent schemes

### Coding Theory: the unit-derived methodology

The unit-derived method in coding theory is shown to be a unique optimal
scheme for constructing and analysing codes. In many cases efficient and
practical decoding methods are produced. Codes with efficient decoding
algorithms at maximal distances possible are derived from unit schemes. In
particular unit-derived codes from Vandermonde or Fourier matrices are
particularly commendable giving rise to mds codes of varying rates with
practical and efficient decoding algorithms. For a given rate and given error
correction capability, explicit codes with efficient error correcting
algorithms are designed to these specifications. An explicit constructive proof
with an efficient decoding algorithm is given for Shannon's theorem. For a
given finite field, codes are constructed which are `optimal' for this field

### Paraunitary Matrices

Design methods for paraunitary matrices from complete orthogonal sets of
idempotents and related matrix structures are presented. These include
techniques for designing non-separable multidimensional paraunitary matrices.
Properties of the structures are obtained and proofs given. Paraunitary
matrices play a central role in signal processing, in particular in the areas
of filterbanks and wavelets

### Group ring cryptography

Cryptographic systems are derived using units in group rings. Combinations of
types of units in group rings give units not of any particular type. This
includes cases of taking powers of units and products of such powers and adds
the complexity of the {\em discrete logarithm} problem to the system.
The method enables encryption and (error-correcting) coding to be combined
within one system. These group ring cryptographic systems may be combined in a
neat way with existing cryptographic systems, such as RSA, and a combination
has the combined strength of both systems. Examples are given.Comment: This is to appear in Intl. J. Pure and Appl. Mat

### On the lower central factors of groups

A general method for calculating or constructing lower central factors of
groups is presented. {\it Relative basic commutators} are defined

### Solving underdetermined systems with error-correcting codes

In an underdetermined system of equations $Ax=y$, where $A$ is an $m\times n$
matrix, only $u$ of the entries of $y$ with $u < m$ are known. Thus $E_jw$,
called `measurements', are known for certain $j\in J \subset
\{0,1,\ldots,m-1\}$ where $\{E_i, i=0,1,\ldots, m-1\}$ are the rows of $A$ and
$|J|=u$. It is required, if possible, to solve the system uniquely when $x$ has
at most $t$ non-zero entries with $u\geq 2t$.
Here such systems are considered from an error-correcting coding point of
view. The unknown $x$ can be shown to be the error vector of a code subject to
certain conditions on the rows of the matrix $A$. This reduces the problem to
finding a suitable decoding algorithm which then finds $x$.
Decoding workable algorithms are shown to exist, from which the unknown $x$
may be determined, in cases where the known $2t$ values are evenly spaced (that
is, when the elements of $J$ are in arithmetic progression) for classes of
matrices satisfying certain row properties. These cases include Fourier
$n\times n$ matrices where the arithmetic difference $k$ satisfies
$\gcd(n,k)=1$, and classes of Vandermonde matrices $V(x_1,x_2,\ldots,x_n)$
(with $x_i\neq 0$) with arithmetic difference $k$ where the ratios $x_i/x_j$
for $i\neq j$ are not $k^{th}$ roots of unity. The decoding algorithm has
complexity $O(nt)$ and in some cases, including the Fourier matrix cases, the
complexity is $O(t^2)$.
Matrices which have the property that the determinant of any square submatrix
is non-zero are of particular interest. Randomly choosing rows of such matrices
can then give $t$ error-correcting pairs to generate a `measuring' code
$C^\perp=\{E_j | j\in J\}$ with a decoding algorithm which finds $x$.
This has applications to signal processing and compressed sensing

### Convolutional codes from unit schemes

Convolutional codes are constructed, designed and analysed using row and/or
block structures of unit algebraic schemes. Infinite series of such codes and
of codes with specific properties are derived. Properties are shown
algebraically and algebraic decoding methods are derived. For a given rate and
given error-correction capability at each component, convolutional codes with
these specifications and with efficient decoding algorithms are constructed.
Explicit prototype examples are given but in general large lengths and large
error capability are achievable. Convolutional codes with efficient decoding
algorithms at or near the maximum free distances attainable for the parameters
are constructible. Unit memory convolutional codes of maximum possible free
distance are designed with practical algebraic decoding algorithms.
LDPC (low density parity check) convolutional codes with efficient decoding
schemes are constructed and analysed by the methods. Self-dual and
dual-containing convolutional codes may also be designed by the methods;
dual-containing codes enables the construction of quantum codes.Comment: This version has substantive changes from previous version

### Free structure of factors

Factors $\frac{X}{Y}$ in a free group $F$ with $Y$ normal in $X$ are
considered. Precise results on the free structure of ${Y}$ relative to the free
structure of ${X}$ when $\frac{X}{Y}$ is abelian are obtained. Some extensions
and applications are given, as for example to the construction of lower central
factors in general groups. A collecting process on free generators, which gives
basic commutator-type free generators for some subgroups, is also presented.
The notion of {\em relative basic commutators} is developed.Comment: 11 page

### Self-dual, dual-containing and related quantum codes from group rings

Classes of self-dual codes and dual-containing codes are constructed. The
codes are obtained within group rings and, using an isomorphism between group
rings and matrices, equivalent codes are obtained in matrix form. Distances and
other properties are derived by working within the group ring. Quantum codes
are constructed from the dual-containing codes

### Linear complementary dual, maximum distance separable codes

Linear complementary dual (LCD) maximum distance separable (MDS) codes are
constructed to given specifications. For given $n$ and $r<n$, with $n$ or $r$
(or both) odd, MDS LCD $(n,r)$ codes are constructed over finite fields whose
characteristic does not divide $n$. Series of LCD MDS codes are constructed to
required rate and required error-correcting capability. Given the field $GF(q)$
and $n/(q-1)$, LCD MDS codes of length $n$ and dimension $r$ are explicitly
constructed over $GF(q)$ for all $r<n$ when $n$ is odd and for all odd $r<n$
when $n$ is even. For given dimension and given error-correcting capability LCD
MDS codes are constructed to these specifications with smallest possible
length. Series of asymptotically good LCD MDS codes are explicitly constructed.
Efficient encoding and decoding algorithms exist for all the constructed codes.
Linear complementary dual codes have importance in data storage,
communications' systems and security.Comment: Small changes from previous versio

- β¦