arXiv subjects
Publications and source records attributed to Ted Hurley.
Unit derived schemes applied to Hadamard matrices are used to construct and analyse linear block and convolutional codes. Codes are constructed to prescribed types, lengths and rates and multiple series of self-dual, dual-containing, linear complementary dual and quantum error-correcting of both linear block {\em and} convolutional codes are derived.
Codes considered as structures within unit schemes greatly extends the availability of linear block and convolutional codes and allows the construction of these codes to required length, rate, distance and type. Properties of a code emanate from properties of the unit from which it was derived. Orthogonal units, units in group rings, Fourier/Vandermonde units and related units are used to construct and analyse linear block and convolutional codes and to construct these to predefined length, rate, distance and type. Self-dual, dual containing, quantum error-correcting and linear complementary dual codes are constructed for both linear block and convolutional codes. Low density parity check linear block and convolutional codes are constructed with no short cycles in the control matrix.
Algebraic methods for the design of series of maximum distance separable (MDS) linear block and convolutional codes to required specifications and types are presented. Algorithms are given to design codes to required rate and required error-correcting capability and required types. Infinite series of block codes with rate approaching a given rational $R$ with $0<R<1$ and relative distance over length approaching $(1-R)$ are designed. These can be designed over fields of given characteristic $p$ or over fields of prime order and can be specified to be of a particular type such as (i) dual-containing under Euclidean inner product, (ii) dual-containing under Hermitian inner product, (iii) quantum error-correcting, (iv) linear complementary dual (LCD). Convolutional codes to required rate and distance and infinite series of convolutional codes with rate approaching a given rational $R$ and distance over length approaching $2(1-R)$ are designed. The designs are algebraic and properties, including distances, are shown algebraically. Algebraic explicit efficient decoding methods are referenced.
Matrices are built and designed by applying procedures from lower order matrices. Matrix tensor products, direct sums or multiplication of matrices are such procedures and a matrix built from these is said to be a {\em separable} matrix. A {\em non-separable} matrix is a matrix which is not separable and is often referred to as {\em an entangled matrix}. The matrices built may retain properties of the lower order matrices or may also acquire new desired properties not inherent in the constituents. Here design methods for non-separable matrices of required types are derived. These can retain properties of lower order matrices or have new desirable properties. Infinite series of required non-separable matrices are constructible by the general methods. Non-separable matrices are required for applications and other uses; they can capture the structure in a unique way and thus perform much better than separable matrices. General new methods are developed with which to construct {\em multidimensional entangled paraunitary matrices}; these have applications for wavelet and filter bank design. The constructions are in addition used to design new systems of non-separable unitary matrices; these have applications in quantum information theory. Some consequences include the design of full diversity constellations of unitary matrices, which are used in MIMO systems, and methods to design infinite series of special types of Hadamard matrices.
Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived since a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank $1$ idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) $U$ is a symmetric unitary matrix if and only if it has the form $I-2E$ for a symmetric idempotent $E$, (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.
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.
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.
General cryptographic schemes are presented where keys can be one-time or ephemeral. Processes for key exchange are derived. Public key cryptographic schemes based on the new systems are easily established. Authentication and signature schemes are implemented. The schemes are an advance on group ring techniques and are easily implemented but highly secure. They may be integrated with error-correcting coding schemes so that encryption/coding and decryption/decoding may be done simultaneously.
The mds (maximum distance separable) conjecture claims that a nontrivial linear mds $[n,k]$ code over the finite field $GF(q)$ satisfies $n \leq (q + 1)$, except when $q$ is even and $k = 3$ or $k = q- 1$ in which case it satisfies $n \leq (q + 2)$. For given field $GF(q)$ and any given $k$, series of mds $[q+1,k]$ codes are constructed. Any $[n,3]$ mds or $[n,n-3]$ mds code over $GF(q)$ must satisfy $n\leq (q+1)$ for $q$ odd and $n\leq (q+2)$ for $q$ even. For even $q$, mds $[q+2,3]$ and mds $[q+2, q-1]$ codes are constructed over $GF(q)$. The codes constructed have efficient encoding and decoding algorithms.
Maximum distance separable (MDS) are constructed to required specifications. The codes are explicitly given over finite fields with efficient encoding and decoding algorithms. Series of such codes over finite fields with ratio of distance to length approaching $(1-R)$ for given $R, \, 0 < R < 1$ are derived. For given rate $R=\frac{r}{n}$, with $p$ not dividing $n$, series of codes over finite fields of characteristic $p$ are constructed such that the ratio of the distance to the length approaches $(1-R)$. For a given field $GF(q)$ MDS codes of the form $(q-1,r)$ are constructed for any $r$. The codes are encompassing, easy to construct with efficient encoding and decoding algorithms of complexity $\max\{O(n\log n), t^2\}$, where $t$ is the error-correcting capability of the code.
Series of maximum distance quantum error-correcting codes are developed and analysed. For a given rate and given error-correction capability, quantum error-correcting codes with these specifications are constructed. The codes are explicit with efficient decoding algorithms. For a given field maximum length quantum codes are constructed.
Entanglement-assisted quantum error-correcting codes (EAQECCs) to desired rate, error-correcting capability and maximum shared entanglement are constructed. Thus for a required rate $R$, required error-correcting capability to correct $t$ errors, mds (maximum distance separable) EAQECCs of the form $[[n,r,d;c]]$ with $R=\frac{r}{n}, d\geq (2t+1), c = (n-r), d= (n-r+1)$ are constructed. Series of such codes may be constructed where the rate and the relative distance approach non-zero constants as $n$ approaches infinity. The codes may also be constructed over prime order fields in which modular arithmetic may be employed.
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.
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.
Orthogonal sets of idempotents are used to design sets of unitary matrices, known as constellations, such that the modulus of the determinant of the difference of any two distinct elements is greater than $0$. It is shown that unitary matrices in general are derived from orthogonal sets of idempotents reducing the design problem to a construction problem of unitary matrices from such sets. The quality of the constellations constructed in this way and the actual differences between the unitary matrices can be determined algebraically from the idempotents used. This has applications to the design of unitary space time constellations.
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.
An isomorphism between the group ring of a finite group and a ring of certain block diagonal matrices is established. The group ring $RG$ of a finite group $G$ is isomorphic to the set of {\em group ring matrices} over $R$. It is shown that for any group ring matrix $A$ of $\mathbb{C} G$ there exists a matrix $P$ (independent of the entries of $A$) such that $P^{-1}AP= \text{diag}(T_1,T_2,\ldots, T_r)$ for block matrices $T_i$ of fixed size $s_i\times s_i$ where $r$ is the number of conjugacy classes of $G$ and $s_i$ are the ranks of the group ring matrices of the primitive idempotents. Using the isomorphism of the group ring to the ring of group ring matrices followed by the mapping $A\mapsto P^{-1}AP$ (where $P$ is of course fixed) gives an isomorphism from the group ring to the ring of such block matrices. Specialising to the group elements gives a faithful representation of the group. Other representations of $G$ may be derived using the blocks in the images of the group elements. Examples are given demonstrating how interesting and useful representations of groups can be derived using the method. For a finite abelian group $Q$ an explicit matrix $P$ is given which diagonalises any group ring matrix of $\mathbb{C} Q$. The matrix $P$ is defined directly in terms of roots of unity depending only on an expression for $Q$ as a product of cyclic groups. The characters and character table of $Q$ may be read off directly from the rows of the diagonalising matrix $P$. This has applications to signal processing and generalises the cyclic case.
Algebraic systems are constructed from which series of maximum distance separable (mds) codes are derived. The methods use unit and idempotent schemes.