Two “Dual” Families of Nearly-Linear Codes over ℤ p , p odd

Since the paper by Hammons e.a. [1], various authors have shown an enormous interest in linear codes over the ring ℤ₄. A special weight function on ℤ₄ was introduced and by means of the so called Gray map ϕ : ℤ₄→ℤ₂ ² a relation was established between linear codes over ℤ₄ and certain interesting non-linear binary codes of even length. Here, we shall generalize these notions to codes over ℤ _p2 where p is an arbitrary prime. To this end, a new weight function will be proposed for ℤ _p2 . Further, properties of linear codes over ℤ _p2 will be discussed and the mapping ϕ will be generalized to an isometry between ℤ _p2 and ℤ _p ^p , resp. between ℤ _p2 ⁿ and ℤ _p ^pn . Some properties of Galois rings over ℤ _q will be described and two dual families of linear codes of length n = p ^m − 1, gcd(m, p) = 1, over ℤ _q will be constructed. Taking q = p ², their images under the new mapping can be viewed as a generalization of the binary Kerdock and the Preparata code, although they miss some of their nice combinatorial properties. Received: June 19, 2000; revised version: November 6, 2000     

The a-invariant of some Reed-Muller Codes

The a-invariant, the defining ideal and the dimension of Reed-Muller codes arising from some subsets of the projective n-space over a finite field are determined. In some cases the minimal distance of the code is given. Received: May 20, 1998; revised version: April 27, 1999     

Reed-Muller Codes on Complete Intersections

By using results and techniques from commutative algebra such as the vanishing ideal of a set of points, its a-invariant, the Hilbert polynomial and series, as well as finite free resolutions and the canonical module, some results about Reed-Muller codes defined on a zero-dimensional complete intersection in the n-projective dimensional space are given. Several examples of this class of codes are presented in order to illustrate the ideas. Received: March 11, 1999; revised version: November 6, 2000     

Generalized GMW Quadriphase Sequences Satisfying the Welch Bound with Equality

This paper gives new families of quadriphase sequences obtained from large linear complexity sequences over Z ₄ by making use of generalized permutation monomials over Galois rings. The construction of these sequences can be seen as a generalization of the binary GMW sequence construction and hence they are referred to as GGMW sequences over Z ₄. The GGMW families satisfy the Welch bound on inner products with equality and it is shown that the root mean square of all the crosscorrelations and out-of-phase autocorrelations (θ_rms), is approximately equal to the quantity ; L being the period of the sequences. However, θ_max, the maximum magnitude of periodic crosscorrelation and out-of-phase autocorrelation, deviates from the optimal value of . Computer results suggest that the number of crosscorrelation values which deviate from the optimal value of is small. The weight structure of these sequences is the same as those of m-sequences over Z ₄. The linear complexity (LC) of the sequences is computed using a generalized Blahuts theorem on the LC of sequences over Z ₄. Received: January 18, 1999; revised version: October 20, 1999     

Intersonic shear crack growth along weak planes

Classical dynamic fracture theories predict the Rayleigh surface wave speed (c _R ) to be the limiting speed of propagation for mode-I cracks in constitutively homogeneous, isotropic, linear elastic materials subjected to remote loading. For mode-II cracks, propagating along prescribed straight line paths, the same theories, while excluding the possibility of crack growth in the speed regime between c _R and the shear wave speed, c _s , do not exclude intersonic (c _s <υ<c _l ) crack tip speeds. In the present study, we provide the first experimental evidence of intersonic crack growth in such constitutively homogeneous and isotropic solids, ever recorded in a laboratory setting. Intersonic shear dominated crack growth, featuring shear shock waves, was observed along weak planes in a brittle polyester resin under far-field asymmetric loading. The shear cracks initially propagate at speeds just above c _s and subsequently accelerate rapidly to the longitudinal wave speed (c _l ) of the solid. At longer times, when steady state conditions are attained, they propagate at speeds slightly higher than √2–c _s . The experimental results compare well with existing asymptotic theories of intersonic crack growth, and the significance of the preferred speed of √2–c _s is discussed. Received: 13 September 1999 / Reviewed and occerted: 19 November 1999     

Hyper-Kloosterman Sums and their Applications to the Coding Theory

The dual code of the Melas code is called the Kloosterman code. The weights of its codewords can be expressed by the Kloosterman sums, and are uniformly distributed with respect to the Sato-Tate measure. In this paper, the hyper-Kloosterman code C _m (q), a generalization of the Kloosterman code is defined, and the uniform distribution property is deduced using the hyper-Kloosterman sums when m is even and p− 1|m. Finally we discuss doubly-evenness for the weights in the binary case. It shows that we can construct infinitely many doubly-even codes in a non-trivial way. Received: October 30, 2000; revised version: February 28, 2001     

Long binary narrow-sense BCH codes are normal

Let C be the binary narrow-sense BCH code of length n = (2 ^m − l)/h, where m is the order of 2 modulo n. Using characters of finite fields and a theorem of Weil, and results of Vladut-Skorobogatov and Lang-Weil we prove that the code C is normal in the non-primitive case h > 1 if 2^m ≥ 4(2th)^{4t + 2}, and in the primitive case h = 1 if m ≥ m ₀ where the constant m ₀ depends only on t.     

Computation of the Dimensions of Symmetry Classes of Tensors Associated with the Finite two Dimensional Projective Special Linear Group

The dimensions of the symmetry classes of tensors associated with the projective special linear group of degree 2 over a field with q elements, PSL ₂(q), are found. Of course we will assume PSL ₂(q) as a subgroup of the symmetric group S _{q +1} because this group has a faithful action on the points of the underlying projective space. We also discuss the non-triviality of the symmetry classes of tensors associated with each irreducible character of PSL ₂(q). Received: October 14, 1998; revised version: November 30, 1999     

Electron‐Deficient and Quinoid Central Unit Engineering for Unfused Ring‐Based A1–D–A2–D–A1‐Type Acceptor Enables High Performance Nonfullerene Polymer Solar Cells with High Voc and PCE Simultaneously

Here, a pair of A₁–D–A₂–D–A₁ unfused ring core‐based nonfullerene small molecule acceptors (NF‐SMAs), BO2FIDT‐4Cl and BT2FIDT‐4Cl is synthesized, which possess the same terminals (A₁) and indacenodithiophene unit (D), coupling with different fluorinated electron‐deficient central unit (difluorobenzoxadiazole or difluorobenzothiadiazole) (A₂). BT2FIDT‐4Cl exhibits a slightly smaller optical bandgap of 1.56 eV, upshifted highest occupied molecular orbital energy levels, much higher electron mobility, and slightly enhanced molecular packing order in neat thin films than that of BO2FIDT‐4Cl . The polymer solar cells (PSCs) based on BT2FIDT‐4Cl:PM7 yield the best power conversion efficiency (PCE) of 12.5% with a V_oc of 0.97 V, which is higher than that of BO2FIDT‐4Cl ‐based devices (PCE of 10.4%). The results demonstrate that the subtle modification of A₂ unit would result in lower trap‐assisted recombination, more favorable morphology features, and more balanced electron and hole mobility in the PM7:BT2FIDT‐4Cl blend films. It is worth mentioning that the PCE of 12.5% is the highest value in nonfused ring NF‐SMA‐based binary PSCs with high V_oc over 0.90 V. These results suggest that appropriate modulation of the quinoid electron‐deficient central unit is an effective approach to construct highly efficient unfused ring NF‐SMAs to boost PCE and V_oc simultaneously.     

Elastic properties of gallium and aluminum nitrides

The Keating model is used to calculate the elastic constants of GaN and AlN for sphalerite and wurtzite structures. The following values of the elastic constants (in gigapascals) were obtained for cubic gallium and aluminum nitride: C ₁₁=325 and 322, C ₁₂=142 and 156, C ₄₄=147 and 138. Pis’ma Zh. Tekh. Fiz. 25, 23–26 (August 12, 1999)