On the weight enumeration of weights less than 2.5 d of Reed—Muller codes

We prove that the families of quaternary Reed – Muller codes obtained by the BQ-Plotkin construction 2009 have bases of minimum weight codewords. In 2020 we found that the quaternary Reed – Muller codes constructed by the quaternary Plotkin approach have the minimum weight bases. Combining these two constructions we prove that all known quaternary linear Reed – Muller codes have bases of minimum weight codewords. The bases are obtained iteratively.

Polynomial-time algorithms for quadratic isomorphism of polynomials: The regular case

Let <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha_{n}</tex> denote the number of cosets with minimum weight <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2^{m}, m + 1)</tex> Reed-Muller code. The <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha_{n}</tex> for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{m-2} \leq n &lt; 2^{m-2} + 2^{m - 4}</tex> is determined.

The following theorem is proved. Let f(x_1,\cdots, x_m) be a binary nonzero polynomial of m variables of degree \nu . H the number of binary m -tuples (a_1,\cdots, a_m) with f(a_1, \cdots, a_m) = 1 is less than 2^{m-\nu+1} , then f can be reduced by an invertible affme transformation of its variables to one of the following forms. \begin{equation} f = y_1 \cdots y_{\nu - \mu} (y_{\nu-\mu+1} \cdots y_{\nu} + y_{\nu+1} \cdots y_{\nu+\mu}), \end{equation} where m \geq \nu+\mu and \nu \geq \mu \geq 3 . \begin{equation} f = y_1 \cdots y_{\nu-2}(y_{\nu-1} y_{\nu} + y_{\nu+1} y_{\nu+2} + \cdots + y_{\nu+2\mu -3} y_{\nu+2\mu-2}), \end{equation} This theorem completely characterizes the codewords of the \nu th-order Reed-Muller code whose weights are less than twice the minimum weight and leads to the weight enumerators for those codewords. These weight formulas are extensions of Berlekamp and Sloane's results.

A fast method to compute the minimum Lee weight and the symmetrized weight enumerator of extended quadratic residue codes (XQR-codes) over the ring Z4 is developed. Our approach is based on the classical Brouwer-Zimmermann algorithm and additionally takes advantage of the large group of automorphisms and the self-duality of the Z4-linear XQR-codes as well as the projection to the binary XQR-codes. As a result, the hitherto unknown minimum Lee distances of all Z4-linear XQR-codes of lengths between 72 and 104 and the minimum Euclidean distances for the lengths 72, 80, and 104 are computed. It turns out that the binary Gray image of the Z4-linear XQR-codes of lengths 80 and 104 has higher minimum distance than any known linear binary code of equal length and cardinality. Furthermore, the Z4-linear XQR-code of length 80 is a new example of an extremal Z4-linear typeII code. Additionally, we give the symmetrized weight enumerator of the Z4-linear XQR-codes of lengths 72 and 80, and we correct the weight enumerators of the Z4-linear XQR-code of length 48 given by Pless and Qian and Bonnecaze et al.

In the design of industrial products ranging from hearing aids to automobiles and aeroplanes, material is distributed so as to maximize the performance and minimize the cost. Historically, human intuition and insight have driven the evolution of mechanical design, recently assisted by computer-aided design approaches. The computer-aided approach known as topology optimization enables unrestricted design freedom and shows great promise with regard to weight savings, but its applicability has so far been limited to the design of single components or simple structures, owing to the resolution limits of current optimization methods. Here we report a computational morphogenesis tool, implemented on a supercomputer, that produces designs with giga-voxel resolution-more than two orders of magnitude higher than previously reported. Such resolution provides insights into the optimal distribution of material within a structure that were hitherto unachievable owing to the challenges of scaling up existing modelling and optimization frameworks. As an example, we apply the tool to the design of the internal structure of a full-scale aeroplane wing. The optimized full-wing design has unprecedented structural detail at length scales ranging from tens of metres to millimetres and, intriguingly, shows remarkable similarity to naturally occurring bone structures in, for example, bird beaks. We estimate that our optimized design corresponds to a reduction in mass of 2-5 per cent compared to currently used aeroplane wing designs, which translates into a reduction in fuel consumption of about 40-200 tonnes per year per aeroplane. Our morphogenesis process is generally applicable, not only to mechanical design, but also to flow systems, antennas, nano-optics and micro-systems.

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, a natural question to ask is which optimal linear codes have an efficient decoding. We show that two binary optimal [ 36 , 19 , 8 ] linear codes and two binary optimal [ 40 , 22 , 8 ] codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes. We project them onto the much shorter length linear [ 9 , 5 , 4 ] and [ 10 , 6 , 4 ] codes over G F ( 4 ) , respectively. This decoding algorithm, called projection decoding, can correct errors of weight up to 3. These [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes respectively have more codewords than any optimal self-dual [ 36 , 18 , 8 ] and [ 40 , 20 , 8 ] codes for given length and minimum weight, implying that these codes are more practical.

The characteristics of the conventional design process, implying repeated analysis of a structure and resizing of its members until a satisfactory design is obtained, is illustrated by means of some simple truss structures. In this process, it is implicitly assumed that by satisfying as closely as possible all requirements placed on the design this will lead to the ‘best’ design. In terms of the maximum stress in the members, this is the well-known principle of the fully stressed design. Effective as this method often is, common situations are identified where this does not lead to an optimum, minimum weight design. Furthermore, the process may be very slowly convergent, in addition to which it offers no help when conditions other than a simple maximum stress apply or, for example, with the optimum shape of a structure. Minimum weight implies economy of material as well as operational savings directly related to reduction in weight. All this provides justification for the formal optimization methods in the remaining chapters of this book. While this chapter is concerned only with truss structures, conclusions reached can, in principle, be taken to apply more widely to the optimization of many other types of structure. A spreadsheet program for the numerical optimization of a simple seven-bar truss provides a first introduction to use of the Solver optimization tool in Microsoft Excel.

The idea to use error-correcting codes in order to construct public key cryptosystems was published in 1978 by McEliece [ME1978]. In his original construction, McEliece used Goppa codes, but various later publications suggested the use of different families of error-correcting codes. The choice of the code has a crucial impact on the security of this type of cryptosystem. Some codes have a structure that can be recovered in polynomial time, thus breaking the cryptosystem completely, while other codes have resisted attempts to cryptanalyze them for a very long time now. In this thesis, we examine different derivatives of the McEliece cryptosystem and study their structural weaknesses. The main results are the following: In chapter 3 we devise an effective structural attack against the McEliece cryptosystem based on algebraic geometry codes defined over elliptic curves. This attack is inspired by an algorithm due to Sidelnikov and Shestakov [SS1992] which solves the corresponding problem for Reed-Solomon codes. The presented algorithm is heuristic polynomial time and thus inverts trapdoors even for very large codes. In chapter 4, we show that the Sidelnikov cryptosystem [S1994], which is based on binary Reed-Muller codes, is insecure. The basic idea of our attack is to use the fact that minimum weight words in a Reed-Muller code have very particular properties. This attack relies on the ability to find minimum weight words in the code, a problem that is, in this specific instance, much easier than general decoding, and feasible for interesting parameters in a modest amount of time. The attack has subexponential running time if the order of the code is kept fixed, and it breaks the large keys as proposed by Sidelnikov in under an hour on a stock PC. In the chapter 5, we finally discuss some of the problems to solve if one attempts to generalize these algorithms.

The tower of the lifting system is the key equipment which ensures the modules being lifted and installed on the deep sea oil extraction platform. As the important bearing structure in the lifting system, the tower has complex force conditions and its quality has great influence on the working performance of the whole lifting system. In terms of the designed gantry tower structure, this paper will use the length and wall thickness of the steel pipe of the gantry tower system as the design variables, the minimum total weight of the tower system as the objective function, the strength and stiffness of the tower system as the constraints. The gantry tower structure is the direct load-bearing structure of the lifting system. Conducting optimization analysis on the gantry tower structure will significantly reduce its weight, make its structure simpler and more reasonable, thus increasing its dependability.

Some construction techniques for self-dual codes are investigated, and the authors construct a singly-even self-dual (48,24,10)-code with a weight enumerator that was not known to be attainable. It is shown that there exists a singly-even self-dual code C' of length n=48 and minimum weight d=10 whose weight enumerator is prescribed in the work of J.H. Conway et al. (see ibid., vol.36, no.5, p.1319-33, 1990). Two self-dual codes of length n are called neighbors, provided their intersection is a code of dimension (n/2)-1. The code C' is a neighbor of the extended quadratic residue code of length 48.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The approach to error correction coding taken by modern digital communication systems started in the late 1940's with the ground breaking work of Shannon, Hamming and Golay. Reed- Muller (RM) codes were an important step beyond the Hamming and Golay codes because they allowed more flexibility in the size of the code word and the number of correctable errors per code word. Whereas the Hamming and Golay codes were specific codes with particular values for q; n; k; and t, the RM codes were a class of binary codes with a wide range of allowable design parameters. Binary Reed-Muller codes are among the most prominent families of codes in coding theory. They have been extensively studied and employed for practical applications. In this research, the performance simulation of Reed-Muller Codec was presented. An introduction on Reed-Muller codes, were introduced that consists of defining the key terms and operation used with the binary numbers. Reed-Muller codes were defined and encoding matrices were discussed. The decoding process was given and some examples were demonstrated to clarify the method. The results and the performance of Reed-Muller encoding were presented and the messages been encoded using the defined matrices were shown. The simulation of the decoding part also been shown. The performance of Reed-Muller codes were then analyzed in terms of its code rate, code length and minimum Hamming distance. The analysis that performed also successfully examines the relationship between the parameters of Reed- Muller coding. The decoding part of the Reed-Muller codes can detect one error and correct it as shown in the examples.

The minimum pseudocodeword weight <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> of a linear graph-based code is more influential in determining decoding performance when decoded via iterative and linear programming decoding algorithms than the classical minimum distance <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> under standard maximum-likelihood decoding Moreover, unlike the minimum distance which is unique to the code regardless of representation, the set of pseudocodewords, and therefore also the minimum pseudocodeword weight, depends on the graph representation used in decoding as well as on the communication channel. This means that a judicious choice of parity-check matrix is crucial for realizing the best potential of any graph-based code. In this paper, we introduce the notion of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pseudoweight</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">redundancy</i> for the memoryless binary symmetric channel (BSC). Analogous to the stopping redundancy in the literature, this parameter gives the smallest number of rows needed for a parity-check matrix to have <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> . We provide some upper bounds on the BSC-pseudoweight redundancy and illustrate the concept with some results for Hamming codes, tree-based and finite geometry LDPC codes, Reed-Muller codes and Hadamard codes.

On the enumeration of self-dual codes

We consider the binary images of (8, 5) shortened cyclic codes. The (8, 5) shortened cyclic codes have a variety of choices. We have generated about 30000 sample codes with different weight distributions. Let S/sub w/ denote the set of generated sample codes with minimum weight w. The largest minimum weight of sample codes is 8. Let A/sub w/ denote the number of codewords of weight w of a sample code. In S/sub 7/, the smallest of A/sub 7/ is 10 for 10 sample codes and the second smallest of A/sub 7/ is 11 for six sample codes. In S/sub 7/, the smallest of A/sub 8/ is 728 for a sample code and the second smallest of A/sub 8/ is 729 for a sample code. We have chosen two sample codes from each of S/sub 7/ and S/sub 8/ which have the smallest and the second smallest sums of A/sub w/ for 7/spl les/w/spl les/9 in S/sub 7/ and S/sub 8/, respectively. For the AWGN channel using BPSK signaling, we have made simulation to evaluate the decoding error probabilities by a soft-decision decoding based on ordered statistics for the chosen four sample codes at SNR 2.0 to 5.0. These error probabilities are considerably smaller than the optimum error probabilities for (64, 40) subcodes of (64, 42) Reed-Muller code.