Eimear Byrne

New families of quadratic almost perfect nonlinear trinomials and multinomials

We introduce two new infinite families of APN functions, one on fields of order 2^2^k for k not divisible by 2, and the other on fields of order 2^3^k for k not divisible by 3. The polynomials in the first family have between three and k+2 terms, the second familys polynomials have three terms.

Linear-Programming Decoding of Nonbinary Linear Codes

A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates LP-based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the ldquomaximum-likelihood (ML) certificaterdquo property. It is also shown that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. It is also proved that if the modulator-channel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. Two alternative polytopes for use with LP decoding are studied, and it is shown that for many classes of codes these polytopes yield a complexity advantage for decoding. These polytope representations lead to polynomial-time decoders for a wide variety of classical nonbinary linear codes. LP decoding performance is illustrated for ternary Golay code with ternary phase-shift keying (PSK) modulation over additive white Gaussian noise (AWGN), and in this case it is shown that the performance of the LP decoder is comparable to codeword-error-rate-optimum hard-decision-based decoding. LP decoding is also simulated for medium-length ternary and quaternary low-density parity-check (LDPC) codes with corresponding PSK modulations over AWGN.

A few more quadratic APN functions

We present an infinite family of quadrinomial APN functions on GF(2n) where n is divisible by 3 but not 9. The family contains inequivalent functions, obtained by setting some coefficients equal to 0. We also discuss the inequivalence proof (by computation) which shows that these functions are new.

Gröbner Bases over Galois Rings with an Application to Decoding Alternant Codes

We develop a theory of Grobner bases over Galois rings, following the usual formulation for Grobner bases over finite fields. Our treatment includes a division algorithm, a characterization of Grobner bases, and an extension of Buchberger?s algorithm. One application is towards the problem of decoding alternant codes over Galois rings. To this end we consider the module M= {(a, b) :aS?b modxr} of all solutions to the so-called key equation for alternant codes, where S is a syndrome polynomial. In decoding, a particular solution (?, ?) ?M is sought satisfying certain conditions, and such a solution can be found in a Grobner basis of M. Applying techniques introduced in the first part of this paper, we give an algorithm which returns the required solution.

Determining the nonlinearity of a new family of APN functions

We compute the Walsh spectrum and hence the nonlinearity of a new family of quadratic multi-term APN functions. We show that the distribution of values in the Walsh spectrum of these functions is the same as the Gold function.

NMR with excitation modulated by Frank sequences

Miniaturized NMR is of growing importance in bio-, chemical, and -material sciences. Other than the magnet, bulky components are the radio-frequency power amplifier and the power supply or battery pack. We show that constant flip-angle excitation with phase modulation following a particular type of polyphase perfect sequences results in low peak excitation power at high response peak power. It has ideal power distribution in both the time domain and the frequency domain. A savings in peak excitation power of six orders of magnitude has been realized compared to conventionally pulsed excitation. Among others, the excitation promises to be of use for button-cell operated miniature NMR devices as well as for complying with specific-absorption-rate regulations in high-field medical imaging.

The linear programming bound for codes over finite Frobenius rings

In traditional algebraic coding theory the linear-programming bound is one of the most powerful and restrictive bounds for the existence of both linear and non-linear codes. This article develops a linear-programming bound for block codes on finite Frobenius rings.

Hamming metric decoding of alternant codes over Galois rings

The standard decoding procedure for alternant codes over fields centers on solving a key equation which relates an error locator polynomial and an error evaluator polynomial by a syndrome sequence. We extend this technique to decode alternant codes over Galois rings. We consider the module M={(a, b): as/spl equiv/b mod x/sup r/} of all solutions to the key equation where s is the syndrome polynomial and r, is the number of rows in a parity-check matrix for the code. In decoding we seek a particular solution (/spl Sigma/, /spl Omega/)/spl isin/M which we prove can be found in a Grobner basis for M. We present an iterative algorithm which generates a Grobner basis modulo x/sup k+1/ from a given basis modulo x/sup k/. At the rth step, a Grobner basis for M is found, and the required solution recovered.

On the equivalence of quadratic APN functions

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to the APN Gold function