Algebraic coding theory is a new and rapidly developing subject, popular for its many practical applications and for its fascinatingly rich mathematical structure. This book provides an elementary yet rigorous introduction to the theory of error-correcting codes. Based on courses given by the author over several years to advanced undergraduates and first-year graduated students, this guide includes a large number of exercises, all with solutions, making the book highly suitable for individual study.

A first course in coding theory, by Raymond Hill. Pp 251. £12·50. 1986. ISBN 0-19-853803-0 (Oxford University Press)

Error Detecting and Error Correcting Codes

We transpose the theory of rank metric. Gabidulin codes (8) W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes. Cambridge. Leech's construction is based on the binary Golay code of length. 24. Later on (13) V.Pless, An Introduction to the Theory of Error-Correcting Codes, Wi. Download full text in PDF Opens in a new window. R. Brualdi and V. Pless, On the length of codes with a given covering radius, Preprint. Introduction to the Theory of Functions of One VariableA. Characterization of (2(q + 1) + 2, 2, t, q)-min-hyper in PG(t, q) (t 3, q 5) and its applications to error-correcting codes. of Coding Theory. 5. 1.

http://ieeexplore.ieee.org/iel5/5/31339/01457228.pdf

Introduction to the theory of error-correcting codes

The famous mathematician S. Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). For any fixed integer q, this is a sequence in n with periodicity q. Ramanujan showed that many standard arithmetic functions in the theory of numbers, such as Euler's totient function φ(n) and the Mobius function μ(n), can be expressed as linear combinations of c_q(n), 1 ≤ q ≤ ∞. In the last ten years, Ramanujan sums have aroused some interest in signal processing. There is evidence that these sums can be used to extract periodic components in discrete-time signals. The purpose of this paper and the companion paper (Part II) is to develop this theory in detail. After a brief review of the properties of Ramanujan sums, the paper introduces a subspace called the Ramanujan subspace S_q and studies its properties in detail. For fixed q, the subspace S_q includes an entire family of signals with properties similar to c_q(n). These subspaces have a simple integer basis defined in terms of the Ramanujan sum c_q(n) and its circular shifts. The projection of arbitrary signals onto these subspaces can be calculated using only integer operations. Linear combinations of signals belonging to two or more such subspaces follows certain specific periodicity patterns, which makes it easy to identify periods. In the companion paper (Part II), it is shown that arbitrary finite duration signals can be decomposed into a finite sum of orthogonal projections onto Ramanujan subspaces.

Ramanujan Sums in the Context of Signal Processing—Part I: Fundamentals

The mathematician Ramanujan introduced a summation in 1918, now known as the Ramanujan sum c_q(n). In a companion paper (Part I), properties of Ramanujan sums were reviewed, and Ramanujan subspaces S_q introduced, of which the Ramanujan sum is a member. In this paper, the problem of representing finite duration (FIR) signals based on Ramanujan sums and spaces is considered. First, it is shown that the traditional way to solve for the expansion coefficients in the Ramanujan-sum expansion does not work in the FIR case. Two solutions are then developed. The first one is based on a linear combination of the first N Ramanujan-sums (with N being the length of the signal). The second solution is based on Ramanujan subspaces. With q_1, q_2,..., q_K denoting the divisors of N; it is shown that x(n) can be written as a sum of K signals x_(qi) (n) ∈ S_(qi). Furthermore, the i_(th) signal x_(qi) (n) has period q_i, and any pair of these periodic components is orthogonal. The components x_(qi) (n) can be calculated as orthogonal projections of x(n) onto Ramanujan spaces S_(qi). Then, the Ramanujan Periodic Transform (RPT) is defined based on this, and is useful to identify hidden periodicities. It is shown that the projection matrices (which compute x_(qi) (n) from x(n)) are integer matrices except for an overall scale factor. The calculation of projections is therefore rendered easy. To estimate internal periods N_∞ <; N of x(n), one only needs to know which projection energies are nonzero.

Ramanujan Sums in the Context of Signal Processing—Part II: FIR Representations and Applications

The Wiener{Khintchine formula plays a central role in statistical mechanics. It is shown here that the problem of prime pairs is related to autocorrelation and hence to a Wiener{Khintchine formula. \Experimental" evidence is given for this. c 1999 Elsevier Science B.V. All rights reserved.

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Ramanujan{Fourier series, the Wiener{Khintchine formula and the distribution of prime pairs

Elliptic Curves with Complex Multiplication and a Character Sum

Given two arithmetical functions f, g, we derive, under suitable conditions, asymptotic formulas for the convolution sums ∑n≤Nf (n)g(n + h) for a fixed number h. To this end, we develop the theory of Ramanujan expansions for arithmetical functions. Our results give new proofs of some old results of Ingham proved by him in 1927 using different techniques.

Ramanujan — Fourier series and a theorem of Ingham

Currently the circle and the sieve methods are the key tools in analytic number theory. In this paper the unifying theme of the two methods is shown to be the Ramanujan - Fourier series.

Linking the Circle and the Sieve: Ramanujan - Fourier Series

The entropy of an orthogonal matrix is defined. It provides a new interpretation of Hadamard matrices as those that saturate the bound for entropy. It appears to be a useful Morse function on the group manifold. It has sharp maximum and other saddle points. The matrices corresponding to the maxima for three and five dimensions are presented. They are integer matrices (up to a rescaling).

R. Padma

Papers

Ramanujan{Fourier series, the Wiener{Khintchine formula and the distribution of prime pairs

Elliptic Curves with Complex Multiplication and a Character Sum

Ramanujan — Fourier series and a theorem of Ingham

Linking the Circle and the Sieve: Ramanujan - Fourier Series

Entropy and Hadamard matrices