Bounds for Coding Theory over Rings.

Coding theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. It has been well established that, with the generalization of the algebraic structure to rings, there is a need to also generalize the underlying metric beyond the usual Hamming weight used in traditional coding theory over finite fields. This paper introduces a generalization of the weight introduced by Shi, Wu and Krotov, called overweight. Additionally, this weight can be seen as a generalization of the Lee weight on the integers modulo 4 and as a generalization of Krotov's weight over the integers modulo 2s for any positive integer s. For this weight, we provide a number of well-known bounds, including a Singleton bound, a Plotkin bound, a sphere-packing bound and a Gilbert-Varshamov bound. In addition to the overweight, we also study a well-known metric on finite rings, namely the homogeneous metric, which also extends the Lee metric over the integers modulo 4 and is thus heavily connected to the overweight. We provide a new bound that has been missing in the literature for homogeneous metric, namely the Johnson bound. To prove this bound, we use an upper estimate on the sum of the distances of all distinct codewords that depends only on the length, the average weight and the maximum weight of a codeword. An effective such bound is not known for the overweight.

Selective magnetic resonance signal suppression by colored Frank excitation.

Inspired by the growing interest in miniaturized NMR devices and their applications in material science as well as in chemical and biological research, low power rf excitation is explored. 1H NMR spectra have been measured with low power Frank excitation and are compared to spectra obtained by single-pulse excitation. Frank excitation consists of a large number of phase-modulated, constant-amplitude rf-pulses. A Frank sequence is divided into packages of discrete phase wavelets that correspond to a scan across a spectral frequency range. The largely coherent excitation is found experimentally to require less power than white noise excitation. The package structure suggests that individual wavelets can be omitted to skip individual frequency regions in the excitation, converting the white Frank excitation into colored Frank excitation. This work explores different approaches of colored, selective Frank excitation for spectroscopy and imaging. It is motivated by the aim to eliminate the rf amplifier from the NMR spectrometer so as to enable further miniaturization of NMR instruments. Colored Frank excitation bears promise as a low-power modality for solvent signal suppression in spectroscopy and motion tagging in magnetic resonance imaging.

A new RF tagging pulse based on the Frank poly-phase perfect sequence.

Radio frequency (RF) spectrally selective multiband pulses or tagging pulses, are applicable in a broad range of magnetic resonance methods. We demonstrate through simulations and experiments a new phase-modulation-only RF pulse for RF tagging based on the Frank poly-phase perfect sequence. In addition, we introduce an extended version with a WURST modulation (Frank-WURST). The new pulses exhibit interesting and flexible spin tagging properties and are easily implemented in existing MR sequences, where they can substitute slice-selective pulses with no additional alterations.

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.