Hadamard and conference matrices

Consider the problem of estimating unbiasedly a linear function, $\lambda?' \omega, \omega' = (\omega_1, \cdots, \omega_p)$, of $p$ weights $\omega_1, \cdots, \omega_p (0 < \omega_i < \infty$ for all $i = 1, \cdots, p)$ when the number of possible weighing operations, $n$, is smaller than $p$. Assume further that the weighing design is of the chemical type (each entry of the design matrix can assume one of the values $-1, 0, +1)$. A weighing design in which the number of weighing operations is smaller than the number of objects, $n < p$, will be called a fractional weighing design. A fractional weighing design is singular. As is well known, if a non-randomized weighing design is singular, one cannot estimate unbiasedly each of the $p$ weights. Some linear functions $\lambda?'\omega$ can be estimated, however, unbiasedly. For example, the sum of all $p$ weights can be estimated unbiasedly by a singular weighing design in which, at each weighing operation, all the objects are placed on one pan. Such a singular weighing design is in fact optimal. One can not attain the same precision in estimating the sum of all the $p$ weights by any non-singular design (see K. S. Banerjee [1]). Some linear functions of the weights can thus be estimated, with a sufficient degree of precision, on the basis of a number of weighing operations, $n$, smaller than the number of weights, $p$. A. M. Mood [4] has shown that if the design matrix of a non-singular weighing design is a Hadamard matrix (if it exists) then the design is optimal. We shall therefore study fractional weighing designs based on Hadamard matrices. If $p$ is such that no Hadamard matrix of order $p$ exists, we shall consider the smallest integer $p'$ larger than $p$, for which a Hadamard matrix exists and add $(p' - p)$ dummy weights. The linear functions $\lambda?'\omega$ can be extended in a way that assigns the dummy weights the coefficient zero. We shall then consider randomized fractional weighing designs which are constructed by choosing at random $n$ rows, independently and with replacement, from the given Hadamard matrix according to a probability vector $\xi$ of order $p$. Every such row specifies a weighing operation to be performed. Non-randomized fractional designs are special cases of the randomized designs under consideration. It is well known that if the weighing design is fractional $(n < p)$ then not all the linear functions $\lambda?'\omega$ are estimable under non-randomized designs. It is shown in the present paper that any linear function $\lambda?'\omega$ can be estimated unbiasedly under a proper randomization scheme. Optimal randomization procedures depend on the functional $\lambda?$. Complete class of unbiased estimation procedures is given for any linear function $\lambda?'\omega$. It is shown that every functional $\lambda?$ specifies a subset of, say $r (1 \leqq r \leqq p)$, admissible weighing operations (rows of the Hadamard matrix); in the sense that if other weighing operations are chosen then, either the estimation procedure is biased, or has a variance larger than the one that can be attained under the admissible weighing operations. It is also proven that if each of the $r$ admissible weighing operations $(1 \leqq r \leqq p)$ is chosen with probability equal to $1/r$, then the corresponding unbiased estimator has a uniformly (in $\omega$ and $\sigma^2$) minimum variance, relative to all the above specified randomization procedures. The formulae of the unbiased estimator of $\lambda?'\omega$ and its variance are extended in Section 4 to the case of random choice without replacement, of $n$ rows out of the $r$ admissible ones. This randomization procedure improves the results that can be obtained under the best randomization procedure with replacement. Finally, the relationship of the present study to the studies of S. Ehrenfeld and S. Zacks [2], [3] is discussed in Section 5.

The multilevel (n-ary) Hadamard matrices, which are related to orthogonal matrices or orthogonal designs, are investigated in this paper. It is assumed that all matrix elements are integers, which make the binary Hadamard matrices as special cases of multilevel (n-ary) Hadamard matrices. It is shown that the multilevel Hadamard matrices of odd order do exist if only two different matrix elements are contained; but, except an unknown case, multilevel (n-ary) Hadamard matrices of odd order do not exist when all matrix elements are successive integers.

Cellular automata and finite fields

This paper's goal is to determine which complete theories of perfect, e-free Ax fields are model-complete. A field K is e-free for a positive integer e if the Galois group g(KS∣K), where Ks is the separable closure of K, is an e-free, profinite group. A perfect field K is pseudo-algebraically closed if each nonvoid, absolutely irreducible variety defined over K has a K-rational point. A perfect, pseudo-algebraically closed field is called an Ax field. The main theorem isA complete theory of e-free Ax fields is model-complete if and only if its field of absolute numbers is e-free.The sufficiency of the latter condition is an easy consequence of a result of Moshe Jarden and Ursel Kiehne [10] and has been noted independently by A. Macintyre and K. McKenna and undoubtedly by others as well. Consequently the necessity of the latter condition is the interesting part of this paper.James Ax [3] initiated the investigation of 1-free Ax fields. He proved that these fields, which he called pseudo-finite fields, are precisely the infinite models of the theory of finite fields. He [3] also presented examples of perfect, 1-free fields which are not pseudo-algebraically closed and an example of a 1-free Ax field whose complete theory is not model-complete. Moshe Jarden [5] showed that the first examples are isolated cases in that almost all, perfect, 1-free, algebraic extensions of a denumerable, Hilbertian field are pseudo-algebraically closed. The results in this paper show that the second example is also an isolated case in that almost all complete theories of 1-free Ax fields are model-complete.

A C-matrix is a square matrix of order m + 1 which is 0 on the main diagonal, has ±1 entries elsewhere and satisfies . Thus, if , I + C is an Hadamard matrix of skew type [3; 6] and, if , iI + C is a (symmetric) complex Hadamard matrix [4]. For m &gt; 1, we must have . Such matrices arise from the quadratic character χ in a finite field, when m is an odd prime power, as [χ(ai – aj)] suitably bordered, and also from some other constructions, in particular those of skew type Hadamard matrices. (For we must have m = a2 + b2, a, b integers.)

The combinatorial power of the companion matrix

Hadamard matrix is a square matrix where any components are -1 or +1, and where any pairs of rows are mutually orthogonal. On the other hand, Hadamard-type matrix on finite fields is a similar one, but has multi-valued components on finite fields. To be more specific, we consider n × n matrices H that have their elements on the given finite fields GF(p), and satisfy HHT = nI under modulo p, where I is an identity matrix. Any additions and multiplications should be executed under modulo p. In the authors' previous studies, some properties of Hadamard-type matrices on finite fields have been proven. For example, it has been shown that the order of a Hadamard-type matrix of odd order on GF(p) is limited to a quadratic residue of a given prime p. On the other hand, it is not clear how many and how various Hadamard-type matrices on GF(p) exist in general. In this paper, we count all possible Hadamard-type matrices on finite fields when the order n of a matrix is small. We also categorize Hadamard-type matrices into six different types when n = 3 and p = 11. In addition, we prove that for any prime p, a Hadamard-type matrix over GF(p) in each of the six types always exists if and only if the order ‘3’ is a quadratic residue of a given p.

Dalam dunia elektronik dan digital, informasi dapat dengan mudahditransfer melalui saluran komunikasi. Pada data yang ditransfer galatdapat muncul dikarenakan oleh berbagai akibat. Untuk menghindarimasalah ini diperlukan kode perbaikan-galat bersama aplikasinya. Konsepdari kode perbaikan-galat adalah untuk menambahkan bit-tambahanpada data agar disaat pengiriman, data tersebut lebih kuat dalammenghadapi gangguan yang hadir di saluran komunikasi. Random ParityCode (RPC) yang dikemukakan oleh Hershey dan Tiemann (1996) adalahsalah satu dari kode yang dimaksud. Artikel ini menunjukan pembuatanaplikasi kode perbaikan-galat yang dibuat berdasarkan konsep RPCdengan bantuan teori Lapangan Terbatas dan Matriks Hadamard. Aplikasidibuat menggunakan Metode Rapid Application Development (RAD).Aplikasi dihasilkan dalam bentuk perangkat lunak komputer. Perangkatlunak tersebut menjadi lebih efisien dengan menerapkan konsepalgoritma “Divide and Conquer”.

Generalizing pairs of complementary sequences and a construction of combinatorial structures

A survey of complex generalized weighing matrices and a construction of quantum error-correcting codes

A set of unimodular code vectors is complementary if the sum, over all code vectors, of the aperiodic autocorrelation sidelobes is zero. Complementary code sets are constructed using a matrix formulation in which code vectors form the columns of a matrix, called a complementary code matrix (CCM). Known construction methods for Hadamard matrices are examined and found to apply to the larger class of CCMs, in some cases with the addition or strengthening of conditions. These constructions include the Sylvester, Williamson, and Kronecker product constructions. Additional approaches for creating new CCMs from available CCMs are introduced and discussed. A future paper will focus on existence results for binary CCMs and on parametric families of unimodular CCMs. Special cases of CCMs are binary CCMs and Hadamard matrices.

In 1952, Perron showed that quadratic residues in a field of prime order satisfy certain additive properties. This result has been generalized in different directions, and our contribution is to provide a further generalization concerning multiplicative quadratic and cubic characters over any finite field. In particular, recalling that a character partitions the multiplicative group of the field into cosets with respect to its kernel, we will derive the number of representations of an element as a sum of two elements belonging to two given cosets. These numbers are then related to the equations satisfied by the polynomial characteristic functions of the cosets. Further, we show a connection, a quasi-duality, with the problem of determining how many elements can be added to each element of a subset of a coset in such a way as to obtain elements still belonging to a subset of a coset.

MDS matrices are used in the design of diffusion layers in many block ciphers and hash functions due to their optimal branch number. But MDS matrices, in general, have costly implementations. So in search for efficiently implementable MDS matrices, there have been many proposals. In particular, circulant, Hadamard, and recursive MDS matrices from companion matrices have been widely studied. In a recent work, recursive MDS matrices from sparse DSI matrices are studied, which are of interest due to their low fixed cost in hardware implementation. In this paper, we present results on the exhaustive search for (recursive) MDS matrices over GL(4, F2). Specifically, circulant MDS matrices of order 4, 5, 6, 7, 8; Hadamard MDS matrices of order 4, 8; recursive MDS matrices from companion matrices of order 4; recursive MDS matrices from sparse DSI matrices of order 4, 5, 6, 7, 8 are considered. It is to be noted that the exhaustive search is impractical with a naive approach. We first use some linear algebra tools to restrict the search to a smaller domain and then apply some space-time trade-off techniques to get the solutions. From the set of solutions in the restricted domain, one can easily generate all the solutions in the full domain. From the experimental results, we can see the (non) existence of (involutory) MDS matrices for the choices mentioned above. In particular, over GL(4, F2), we provide companion matrices of order 4 that yield involutory MDS matrices, circulant MDS matrices of order 8, and establish the nonexistence of involutory circulant MDS matrices of order 6, 8, circulant MDS matrices of order 7, sparse DSI matrices of order 4 that yield involutory MDS matrices, and sparse DSI matrices of order 5, 6, 7, 8 that yield MDS matrices. To the best of our knowledge, these results were not known before. For the choices mentioned above, if such MDS matrices exist, we provide base sets of MDS matrices, from which all the MDS matrices with the least cost (with respect to d-XOR and s-XOR counts) can be obtained. We also take this opportunity to present some results on the search for sparse DSI matrices over finite fields that yield MDS matrices. We establish that there is no sparse DSI matrix S of order 8 over F28 such that S8 is MDS.

This book provides the reader with all the tools necessary to implement modern error-processing schemes. It assumes only a basic knowledge of linear algebra and develops the mathematical theory in parallel with the codes. Central to the text are worked examples which motivate and explain the theory. The book is in four parts. The first introduces the basic ideas of coding theory. The second and third parts cover the theory of finite fields and give a detailed treatment of BCH and Reed-Solomon codes. These parts are linked by their use of Euclid's algorithm as a central technique. The fourth part is devoted to Goppa codes, both classical and geometric, concluding with the Skorobogatov-Vladut error processor. A special feature of this part is a simplified (but rigorous) treatment of the geometry of curves.

A field K is regularly closed if every absolutely irreducible affine variety defined over K has K-rational points. This notion was first isolated by Ax [A] in his work on the elementary theory of finite fields. Later Jarden [J2] and Jarden and Kiehne [JK] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field K with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in K. The method of proof employed in [J1], [J2] and [JK] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [CDM] and to the forthcoming book by Fried and Jarden [FJ] for a more thorough discussion of the latest results. We would like to thank Moshe Jarden, Angus Macintyre, and Zoe Chatzidakis for their comments on an earlier version of this paper.A countable field K is ω-free if the absolute Galois group , where is the algebraic closure of K and is the free profinite group on ℵ0 generators.