On algebraic decoding of the Z4-linear Goethals-likecodes

The Z₄-linear Goethals-like code of length 2^m has 2^2m+1-3m-2 codewords and minimum Lee distance 8 for any odd integer m⩾3. We present an algebraic decoding algorithm for all Z₄-linear Goethals-like codes C_k introduced by Helleseth et al.(1995, 1996). We use Dickson polynomials and their properties to solve the syndrome equations     

On coset weight distributions of the Z4-linear Goethalscodes

We study the coset weight distributions of two well-known families of codes: the three-error-correcting binary Z₄-linear Goethals codes of length N=2^m+1, m⩾3 odd, and the Z₄ -linear Goethals codes over Z₄ of length n=N/2=2^m . The hard case is the weight distributions of cosets of weight 4. To know the weight distribution of the coset of weight 4 we have to know the number of codewords of weight 4 in such a coset. Altogether, there are nine different types of cosets of weight 4. For six cases, we give the exact expressions for the number of codewords of weight 4, and for three other cases, we give such expressions in terms of Kloosterman sums     

An Assmus-Mattson theorem for Z4-codes

The Assmus-Mattson theorem is a method to find designs in linear codes over a finite field. The purpose of this paper is to give an analog of this theorem for Z₄-codes by using the harmonic weight enumerator introduced by Bachoc. This theorem can find some 5-designs in the lifted Golay code over Z₄ which were discovered previously by other methods     

On the weight hierarchy of Preparata codes over Z4

Hammons et al. (see ibid., vol.40, p.301-19, 1994) showed that, when properly defined, the binary nonlinear Preparata code can be considered as the Gray map of a linear code over Z₄, the so called Preparata code over Z₄. We consider the rth generalized Hamming weight d_r(m) of the Preparata code of length 2^m over Z₄. For any m⩾3, d_r(m) is exactly determined for r=0.5, 1, 1.5, 2, 2.5 and 3.0. For a composite m, we give an upper bound on d_r(m) using the lifting technique. For m=3, 4, 5, 6 and 8, the weight hierarchy is completely determined. In the case of m=7, the weight hierarchy is completely determined except for d₄(7)     

Translates of linear codes over Z4

We give a method to compute the complete weight distribution of translates of linear codes over Z₄. The method follows known ideas that have already been used successfully by others for Hamming weight distributions. For the particular case of quaternary Preparata codes, we obtain that the number of distinct complete weights for the dual Preparata codes and the number of distinct complete coset weight enumerators for the Preparata codes are both equal to ten, independent of the code length     

Optimal binary linear codes and Z4-linearity

We give necessary and sufficient conditions for a binary linear code to be Z₄-linear. Especially we treat optimal, binary linear codes and determine all such codes with minimum weight less or equal to six which are Z₄-linear     

Z2k-linear codes

We introduce a generalization to Z₂k of the Gray map and generalized versions of Kerdock and Delsarte-Goethals codes     

Cyclic codes over Z4, locator polynomials, and Newton'sidentities

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock (1972) and Preparata (1968) codes that can be very simply constructed as binary images, under the Gray map, of linear codes over Z₄ that are defined by means of parity checks involving Galois rings. This paper describes how Fourier transforms on Galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes. These methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of Z ₄. Linear codes with length 2^m (m, odd) and size 2(2^m+1-5m-2). The Gray image of the code of length 32 is the best (64, 2³⁷) code that is presently known. This paper also determines the exact minimum Lee distance of the linear codes over Z₄ that are obtained from the extended binary two- and three-error-correcting BCH codes by Hensel lifting. The Gray image of the Hensel lift of the three-error-correcting BCH code of length 32 is the best (64, 2³²) code that is presently known. This code also determines an extremal 32-dimensional even unimodular lattice     

Optimal biphase sequences with large linear complexity derived fromsequences over Z4

New families of biphase sequences of size 2^r-1+1, r being a positive integer, are derived from families of interleaved maximal-length sequences over Z₄ of period 2(2^r-1). These sequences have applications in code-division spread-spectrum multiuser communication systems. The families satisfy the Sidelnikov bound with equality on &thetas;_max, which denotes the maximum magnitude of the periodic cross-correlation and out-of-phase autocorrelation values. One of the families satisfies the Welch bound on &thetas;_max with equality. The linear complexity and the period of all sequences are equal to r(r+3)/2 and 2(2 ^r-1), respectively, with an exception of the single m-sequence which has linear complexity r and period 2^r-1. Sequence imbalance and correlation distributions are also computed     

The Z4-linearity of Kerdock, Preparata, Goethals, andrelated codes

Certain notorious nonlinear binary codes contain more codewords than any known linear code. These include the codes constructed by Nordstrom-Robinson (1967), Kerdock (1972), Preparata (1968), Goethals (1974), and Delsarte-Goethals (1975). It is shown here that all these codes can be very simply constructed as binary images under the Gray map of linear codes over Z₄, the integers mod 4 (although this requires a slight modification of the Preparata and Goethals codes). The construction implies that all these binary codes are distance invariant. Duality in the Z₄ domain implies that the binary images have dual weight distributions. The Kerdock and “Preparata” codes are duals over Z₄-and the Nordstrom-Robinson code is self-dual-which explains why their weight distributions are dual to each other. The Kerdock and “Preparata” codes are Z₄-analogues of first-order Reed-Muller and extended Hamming codes, respectively. All these codes are extended cyclic codes over Z₄, which greatly simplifies encoding and decoding. An algebraic hard-decision decoding algorithm is given for the “Preparata” code and a Hadamard-transform soft-decision decoding algorithm for the I(Kerdock code. Binary first- and second-order Reed-Muller codes are also linear over Z₄ , but extended Hamming codes of length n⩾32 and the Golay code are not. Using Z₄-linearity, a new family of distance regular graphs are constructed on the cosets of the “Preparata” code     

Z4×(F2+uF2)上的一类循环码

定义了Z₄×（F₂+uF₂）上的循环码,明确了一类循环码的生成元结构,给出了该类循环码的极小生成元集.利用Gray映射,构造了一些二元非线性码.     

Further results on generalized Hamming weights for Goethals andPreparata codes over Z4

This article contains results on the generalized Hamming weights (GHW) for the Goethals and Preparata codes over Z₄. We give an upper bound on the rth generalized Hamming weights d_r(m,j) for the Goethals code G_m(j) of length 2^m over Z ₄, when m is odd. We also determine d_3.5(m,j) exactly. The upper bound is shown to be tight up to r=3.5. Furthermore, we determine the rth generalized Hamming weight d_r(m) for the Preparata code of length 2^m over Z₄ when r=3.5 and r=4     

有限环Z4上码字广度的性质及其递归算法

研究码及码字的结构是编码理论的一个重要研究方向.该文定义了环Z4上码字的一种数学特征,即码字的广度.研究了码字广度的一些性质,给出了计算Z4环上码字广度的两种递归算法,并对Z4环上的码字广度与深度之间的关系进行了初步的讨论.     

Algebraic construction of cyclic codes over Z8 with agood Euclidean minimum distance

Let S(8) denote the set of the eight admissible signals of an 8PSK communication system. The alphabet S(8) is endowed with the structure of Z₈, the set of integers taken modulo 8, and codes are defined to be Z₈-submodules of Z₈ⁿ. Three cyclic codes over Z₈ are then constructed. Their length is equal to 6, 8, and 7, and they, respectively, contain 64, 64, and 512 codewords. The square of their Euclidean minimum distance is equal to 8, 16-4√2 and 10-2√2, respectively. The size of the codes of length 6 and 7 can be doubled while the Euclidean minimum distance remains the same     

On Type II codes over F4

Previously, Type II codes over F₄ have been introduced as Euclidean self-dual codes with the property that all Lee weights are divisible by four. In this paper, a number of properties of Type II codes are presented. We construct several extremal Type II codes and a number of extremal Type I codes. It is also shown that there are seven Type II codes of length 12, up to permutation equivalence     

Laser4F3/2→4I9/2and4F3/2→4I13/2transitions in KY(WO4)2:Nd3+

We report the observation of lasing at 0.9137 μm and 1.3545 μm in neodymium-doped KY(WO4)2at 77 K. Transition cross sections, fluorescent line width, and branching ratios are given.     

A new Z11 impedance technique to extract mobility andsheet carrier concentration in HFETs and MESFETs

Conventional techniques to extract channel mobility, μ, and sheet carrier concentration, n_s, in heterostructure field-effect transistors (HFETs) do not account for the distributed nature of the device. This can result in substantial errors. To address this, we have developed a new technique that consists of measuring the gate-to-source impedance with the drain floating (Z₁₁) over a broad frequency range. A transmission line model (TL model) is fitted to Re[Z₁₁], thus obtaining the gate capacitance and channel resistance (and consequently μ(V_GS) and n_s(V _GS)) in a single measurement. We demonstrate this technique in InAlAs-InGaAs on InP HFET's. The TL model faithfully represents Z₁₁ from 100 Hz to 15 MHz. Our technique can easily be automated and thus is a good tool for accurate charge control in an industrial environment     

MBi_4Ti_4O_(15)基陶瓷的研究现状与发展趋势

铋层状结构无铅铁电陶瓷具有良好的抗疲劳性能和较高的居里温度,在铁电存储以及高温压电器件方面具有广阔的应用前景。介绍了MBi4Ti4O15基铋层状陶瓷的结构特点,综述了微量元素掺杂、粉体制备方法和晶粒定向技术对该陶瓷铁电压电性能的影响。并展望了MBi4Ti4O15基铋层状无铅铁电陶瓷未来的发展趋势。     

Single-longitudinal-mode LiNdP4O12laser

Single-longitudinal-mode operation of a LiNdP4O12laser, pumped longitudinally with an argon laser, is reported. Single-frequency operation was obtained with a 0.3-mm-thick crystal using an etalon effect of crystal surfaces without any frequency selective device.     

Initiation of NF3-H2chemical reactions via CO2laser irradiation of NF4BF4

NF4BF4has been irradiated with a low-power CO2laser to produce reactive species which then initiate reactions in an ambient NF3-H2mixture. The laser-induced decomposition of NF4BF4in vacuum was measured as a function of laser power and energy. The laser-induced decomposition threshold was determined to be 40 mJ, which corresponds to a laser fluence of 20 J/cm². This same value was determined for the initiation threshold of NF3-H2reactions via CO2laser irradiation of NF4BF4. Visible and infrared emissions were observed from initiated NF3-H2mixtures. This radiative technique has attractive features for initiating reactions in solid-gas systems.