On minimum distance bounds for abelian codes

This paper is an exposition of two methods of formulating a lower bound for the minimum distance of a code which is an ideal in an abelian group ring. The first, a generalization of the cyclic BCH (Bose-Chaudhuri-Hoquenghem) bound, was proposed by Camion [2]. The second method, presented by Jensen [4], allows the application of the BCH bound or any of its improvements by viewing an abelian code as a direct sum of concatenations of cyclic codes. This second method avoids the mathematical analysis required for a direct generalization of a cyclic bound to the abelian case. It can produce a lower bound that improves the generalized BCH bound. We present simple algorithms for 1) deriving the generalized BCH bound for an abelian code 2) determining direct sum decompositions of an abelian code to concatenated codes and 3) deriving a bound on an abelian code, viewed as a direct sum of concatenated codes, by applying the cyclic BCH bound to the inner and outer code of each concatenation. Finally, we point out the applicability of these methods to codes that are not ideals in abelian group rings.     

Dual codes of systematic group codes over abelian groups

For systematic codes over finite fields the following result is well known: If [I¦P] is the generator matrix then the generator matrix of its dual code is [ ?P ^tr¦I]. The main result is a generalization of this for systematic group codes over finite abelian groups. It is shown that given the endomorphisms which characterize a group code over an abelian group, the endomorphisms which characterize its dual code are identified easily. The self-dual codes are also characterized. It is shown that there are self-dual and MDS group codes over elementary abelian groups which can not be obtained as linear codes over finite fields.     

Performance bounds for uniformly interleaved serial concatenations with a general inner system

Average performance bounds based on uniform interleaving are considered for a serial concatenation consisting of an outer code, an interleaver and an inner coded modulation system. The exact input-output weight spectrum is derived for the case when only the outer code has the uniform error property. Thereby, an upper bound to the bit error probability under maximum likelihood detection is obtained. However, numerous examples for serially concatenated continuous phase modulation show that neglecting the non-uniform error property of the inner system gives an accurate and less complex approximation. Other inner coded modulations may behave differently.     

A generalization of justesen codes

Abstract

By using alternant codes as outer codes in the concatenated structure for Justesen codes, a generalization of Justesen codes which completely meets the Zyablov bound is constructed. For this class of codes, the inner codes are explicitly defined while the outer codes are not.     

Magnetic spherical Couette flow in linear combinations of axial and dipolar fields

We present axisymmetric numerical calculations of the fluid flow induced in a spherical shell with inner sphere rotating and outer sphere stationary. A magnetic field is also imposed, consisting of particular linear combinations of axial and dipolar fields, chosen to make B _r  = 0 at either the outer sphere, or the inner, or in between. This leads to the formation of Shercliff shear layers at these particular locations. We then consider the effect of increasingly large inertial effects and show that an outer Shercliff layer is eventually destabilized, an inner Shercliff layer appears to remain stable, and an in-between Shercliff layer is almost completely disrupted even before the onset of time-dependence, which does eventually occur though.     

Bandwidth efficient concatenated coding schemes

Concatenated coded modulation is a powerful method for constructing coding schemes with large coding gain and low decoding complexity. For the concatenated scheme, the outer code has redundancy and the overall transmission rate is reduced. The problem of bandwidth expansion can be overcome by two procedures. First, the code rate of inner codes is increased with a special coding design. Second, a two-level concatenated scheme is constructed to compensate the rate loss. This way, the low-complexity concatenated coded modulation schemes can be designed without bandwidth expansion as compared with the uncoded scheme.     

Packet-LDPC codes for tape drives

In this paper, we introduce packet low-density parity-check (packet-LDPC) codes for high-density tape storage systems. We report on the performance of two error control code (ECC) architectures based on the packet-LDPC codes. The architectures are designed to be (approximately) compatible with the widely used ECMA-319 ECC standard based on two interleaved concatenated 8-bit Reed-Solomon (RS) codes. One architecture employs an inner RS code; the other employs an inner turbo product code with single parity-check constituent codes (TPC-SPC). Both employ a packet-LDPC code as the outer code. As for the ECMA-319 system, both schemes are required to correct noise bursts due to media defects and synchronization loss, as well as the loss of one of eight tracks (due to a head clog, for example). We show that the first packet-LDPC code architecture substantially outperforms the ECMA-319 scheme and is only a few tenths of a decibel inferior to a long, highly complex 12-bit RS scheme. The second architecture outperforms both the ECMA-319 and the long RS code scheme.     

Trends in formation energies and elastic moduli of ternary and quaternary transition metal nitrides

The paper deals with characteristics of a wide range of ternary and quaternary metal nitrides (M = Ti, Zr, Hf, V, Nb or Ta) of various compositions obtained by ab initio calculations. We focus on the formation energies (E _form), bulk moduli (B), shear moduli (G) and a difference of B and G from the weighted average of B and G of binary metal nitrides (?B and ?G). We show numerous monotonous dependencies, and identify exceptions to them. For elastic moduli of M¹M²N we find that ?B decreases (down to ?19 GPa) and ?G increases (up to 20 GPa) with increasing difference between atomic radii of M¹ and M². In parallel, low ?B and high ?G correspond to high E _form and |E _form|, respectively. E _form of M¹M²N increases with increasing difference between atomic radii and electronegativities of M¹ and M². The lowest E _form values were observed for Ta-containing compositions, and the difference between E _form of TaM¹M²N and M¹M²N is more significant for lower atomic radius and higher electronegativity of M¹ and M². Overall, we present trends which allow one to use fundamental arguments (such as atomic radii and electronegativities) to understand and predict which compositions form (nano)composites, which compositions form (stable) solid solutions, and which materials exhibit enhanced elastic moduli. The phenomena shown can be tested experimentally, and examined for even wider range of materials.     

Thin film coatings on capillary inner walls by microplasma

The effect of a strong static magnetic field on a capillary microplasma generated by RF (13.56 MHz) excitation using the parallel-plate electrodes was studied. If the magnetic field is perpendicular to both RF electric field and a capillary axis and also the Larmor radius of an electron r_L is sufficiently smaller than both the electron mean free path λ_e and a discharge gap d_g, the oscillating E × B drift will appear effectively in a parallel direction with a capillary axis. It was found that a magnetized capillary microplasma generates under the condition of r_L < λ_e and r_L < d_g. Next, to examine the effect of a strong magnetic field on thin film deposition, SiO₂ thin film coating on the inner wall of a poly(propylene) (PP) narrow tube using tetraethoxysilane as the precursor was demonstrated under the condition with/without magnetic field of 0.4 T at atmospheric pressure. SEM images and XPS spectra showed that SiO₂ films with a smooth surface were prepared on the tube inner wall in the both cases.     

Rate allocation for serial concatenated convolutional codes

The paper proposes an analytical method to solve the rate allocation problem in serial concatenated convolutional codes (SCCCs). The goal is to find the best rate allocation between the inner and the outer constituent convolutional codes of an SCCC for constant overall code rate, interleaver size and complexity. Simulation results are shown in the paper to demonstrate the optimum and superior design criteria. In addition, the `density evolution' model is shown to verify the proposed rate allocation method, while indicating that a high rate inner code should not be used for SCCCs. Finally, it is also shown that the upper bounds on BER of ML decoded SCCCs do not provide good design criteria for allocating the rate in iteratively decoded SCCCs     

Scaling law in voltage — current characteristics of hard superconductors

From scaling laws for pinning we have suggested scaling laws for voltage — current characteristics of type II superconductors. After linearization of J_c dependence on magnetic field B, temperature T and stress ?, they lead to exponential dependence of the voltage E on these parameters. In order to obtain general scaling laws, then the temperature, field and stress dependence of the denominator J₀in E = E₀ exp [(J?J_c)/J₀] should also be determined.The generalization of the instability criterion leads to the conclusion that the prestrain of the superconducting material can enhance or deteriorate the stability of the winding.     

Application of MTR soft-decision decoding in multiple-head magnetic recording systems

Application of a simple approach for the soft-decision decoding of Maximum-Transition-Run (MTR) codes has been presented in this paper. A lowdemanded hardware realization have been proposed for soft-decision decoding in MTR basic AND, OR and XOR logic circuits. The suggested approach is explored over the two-track, two-head E²PR4 partial response magnetic recording system. The overall two-track channel detection complexity reduction of 41·9% is offered in simulation scheme, encoded by Low-Density Parity-Check (LDPC) code, serially concatenated with inner MTR. The 1·9 dB coding gain has been obtained, comparing to uncoded channel and assuming the AWGN noise presence.     

Dependence of energy confinement time on toroidal magnetic field in the TUMAN-3M tokamak

The influence of toroidal magnetic field on the energy confinement time ??_E in the ohmic H-mode has been studied in the TUMAN-3M tokamak with low magnetic field. The experiments were performed at a toroidal magnetic field of B _T = 0.68?1.0 T, which is about twice as large as that (0.25?C0.5 T) studied in analogous experiments on the NSTX and MAST spherical tokamaks. The results are indicative of a strong dependence of the energy confinement time on toroidal magnetic field: ??_E ?? B _T ^0.75?C0.8 . This scaling is much stronger than that projected for the ITER (??_{E_IPB98} ?? B _T ^0.15 ), while being somewhat weaker than the scalings observed on the NSTX and MAST devices. The stronger (as compared to the ITER scaling) dependence of ??_E on B _T observed in these experiments should be taken into account in designing thermonuclear facilities with small aspect ratios and toroidal magnetic fields??in particular, fusion neutron sources.     

Structure of codes over the ring Z_{3}[v]/\langle v^{3}-v\rangle

In this paper, we study the structure of linear codes over the non chain ring $Z_{3}[v]/\langle v^{3}-v\rangle $ . In order to study the codes, we first study the structure of this ring via a distance preserving Gray map which also induces a relation between codes over this ring and ternary codes. Further, the algebraic structure of cyclic and dual codes is also studied. A MacWilliams type Identity between the Gray weight enumerators of the original code and its dual is established. In all cases examples that illustrate the theorems and lemmas are provided. Also, a BCH-type bound and an example that attains this bound is presented.     

Large-Area YBa2Cu3O7 − x Films on Sapphire with Excellent Microwave Power Handling Capability

Crack-free thick YBa₂Cu₃O_{7 ? x} films are prepared on CeO₂ buffered r-cut sapphire (2 inch in diameter) with thickness up to 700 nm, smooth surfaces (“peak-to-valley” roughness <10 nm), high critical currents (J _C > 2 MA/cm² at 77 K and 0 T), and low microwave surface resistances (R _s(77K) ?.4mω and R _s(4.2K) ?.110μω at 19.15GHz) comparable to the best values reported in the literature for YBCO films on structurally better matched substrates. These thick YBCO films were able to handle high microwave power corresponding to magnetic field amplitudes (B _HF) up to 54, 37, and 17.4 mT at 4.2, 50, and 77 K, respectively, which for the lower temperatures were limited by the available power of the 25-W HF amplifier. The high-power performance, which to our knowledge belongs to the best reported so far for unpatterned YBa₂Cu₃O_{7 ? x} films, was achieved without any degradation of the samples despite frequent thermal cycling.     

Double power law for basic creep of concrete

The dependence of creep on load duration (t?t′) as well as age at loading t′ is described by the law [1+? ₁ t′^-m (t-t′) ⁿ ]/E ₀ in which m, n,? ₁ E₀=material parameters which are determined from test data by optimization techniques. The law is limited to basic creep, but with different values of material parameters it can also describe drying creep up to a certain time. The previous formulations are extended by introducing the age dependence. This also enhances the reliability in long-term extrapolation of creep data. Substituting t?t′=0.001 day, the law also yields the correct age dependance of the conventional elastic modulus, E. If E₀, which is much larger than E, were replaced by E (as implied by previous power laws without age dependence), the age dependence of creep curves obtained by data analysis would be more scattered, the age dependence of E would have to be described by a separate formula, and more material parameters would be necessary to fit test data. The simplicity of the double power law is a major advantage for statistical evaluation of test data.     

Knight Shift Data and Superconducting T c Modeling Based on an Effective Hole Model

The superconducting T _c of cuprates have been modeled on a linear scaling with hole concentration per CuO₂ plane and a deleterious influence of bond resonance with the apical system (effective hole formalism). In cases where distribution between various hole reservoirs is not trivial, Knight shift can provide actual hole concentrations. It is shown here that when Knight shift data are used in an effective hole algorithm, satisfactory T _c predictions can be made, corroborating the deleterious influence on T _c of apical O and earlier assumptions concerning hole distributions. For the case of stacking of more than two CuO₂ planes, the inner plane has to be treated as an infinite layer analog in the effective hole model. A separation into inner and outer planes with different dopings is indeed observed by Knight shift, with higher doping in the latter. This is explained here as being due to a tendency to equalize an effective doping degree, as the outer planes lose holes in resonance of bonding with the apical system.     

Dual codes of systematic group codes over abelian groups

For systematic codes over finite fields the following result is well known: If [I∣P] is the generator matrix then the generator matrix of its dual code is The main result is a generalization of this for systematic group codes over finite abelian groups. It is shown that given the endomorphisms which characterize a group code over an abelian group, the endomorphisms which characterize its dual code are identified easily. The self-dual codes are also characterized. It is shown that there are self-dual and MDS group codes over elementary abelian groups which can not be obtained as linear codes over finite fields. Received March 7, 1995; revised version April 3, 1996     

Matrix-Product Codes over ? q

Codes C ₁ ,…,C _M of length n over ? _q and an M × N matrix A over ? _q define a matrix-product code C = [C ₁ …C _M ] ·A consisting of all matrix products [c ₁ … c _M ] ·A. This generalizes the (u|u+v)-, (u+v+w|2u+v|u)-, (a+x|b+x|a+b+x)-, (u+v|u-v)- etc. constructions. We study matrix-product codes using Linear Algebra. This provides a basis for a unified analysis of |C|, d(C), the minimum Hamming distance of C, and C ^⊥. It also reveals an interesting connection with MDS codes. We determine |C| when A is non-singular. To underbound d(C), we need A to be `non-singular by columns (NSC)'. We investigate NSC matrices. We show that Generalized Reed-Muller codes are iterative NSC matrix-product codes, generalizing the construction of Reed-Muller codes, as are the ternary `Main Sequence codes'. We obtain a simpler proof of the minimum Hamming distance of such families of codes. If A is square and NSC, C ^⊥ can be described using C ₁ ^⊥, …,C _M ^⊥ and a transformation of A. This yields d(C ^⊥). Finally we show that an NSC matrix-product code is a generalized concatenated code. Received: July 20, 1999; revised version: August 27, 2001