Weight Codewords Research Articles

Stability of k mod p multisets and small weight codewords of the code generated by the lines of PG(2,q)

In this paper, we prove a stability result on k mod p multisets of points in PG(2,q), q=ph. The particular case k=0 is used to describe small weight codewords of the code generated by the lines of PG(2,q), as linear combination of few lines. Earlier results proved this for codewords with weight less than 2.5q, while our result is valid until cqq. It is sharp when 27<q square and h≥4. When q is a prime, De Boeck and Vandendriessche (see [2]) constructed a codeword of weight 3p−3 that is not the linear combination of three lines. We characterise their example.

On explicit minimum weight bases for extended cyclic codes related to Gold functions

Minimum weight bases of some extended cyclic codes can be chosen from the affine orbits of certain explicitly represented minimum weight codewords. We find such bases for the following three classes of codes: the extended primitive 2-error correcting BCH code of length \(n=2^m,\) where \(m\ge 4\) (for \(m\ge 20\) the result was proven in Grigorescu and Kaufman IEEE Trans Inf Theory 58(I. 2):78–81, 2011), the extended cyclic code \(\bar{C}_{1,5}\) of length \(n=2^m,\) odd m, \(m\ge 5,\) and the extended cyclic codes \(\bar{C}_{1,2^i+1}\) of lengths \(n=2^m,\) \((i,\,m)=1\) and \(3\le i\le \frac{m-5}{4}-o(m).\)

On Constructions of Reed-Muller Subcodes

In this letter, subcodes constructed from Reed–Muller codes by removal of generator matrix rows are considered. A new greedy algorithm based on the overlap of generator matrix rows is developed. To select the best subcode generated by the greedy algorithm, the number of minimum weight code words is determined. Computer simulations confirm that the greedy algorithm outperforms the three other construction methods, generating the best codes among all presented subcodes.

Tight Lower and Upper Bounds on the Minimum Distance of LDPC Codes

In this letter, we obtain lower and upper bounds on the minimum distance $d_{\min }$ of low-density parity-check (LDPC) codes. The bounds are derived by categorizing the non-zero code words of an LDPC code into two categories of elementary and non-elementary. The first category contains code words whose induced subgraph has only degree-2 check nodes. We propose an efficient search algorithm that can find the elementary code words of an LDPC code with weight less than a certain value $a_{\max }$ , exhaustively. We also derive a lower bound $L_{ne}$ on the weight of non-elementary code words. By performing the search with $a_{\max } = L_{ne}$ , we either obtain an elementary code word with the smallest weight $d_{\min }$ , or establish the lower bound of $L_{ne}$ on $d_{\min }$ . For the upper bound, we modify our search algorithm to reach elementary codewords of larger weights at the cost of being non-exhaustive. Once such a codeword is found, its weight acts as an upper bound on $d_{\min }$ . We examine a large number of regular and irregular LDPC codes, and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, $d_{\min }$ . Finding $d_{\min }$ , or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm.

Polar Codes for Covert Communications over Asynchronous Discrete Memoryless Channels.

This paper introduces an explicit covert communication code for binary-input asynchronous discrete memoryless channels based on binary polar codes, in which legitimate parties exploit uncertainty created by both the channel noise and the time of transmission to avoid detection by an adversary. The proposed code jointly ensures reliable communication for a legitimate receiver and low probability of detection with respect to the adversary, both observing noisy versions of the codewords. Binary polar codes are used to shape the weight distribution of codewords and ensure that the average weight decays as the block length grows. The performance of the proposed code is severely limited by the speed of polarization, which in turn controls the decay of the average codeword weight with the block length. Although the proposed construction falls largely short of achieving the performance of random codes, it inherits the low-complexity properties of polar codes.

Minimum distance of orthogonal line-Grassmann codes in even characteristic

In this paper we determine the minimum distance of orthogonal line-Grassmann codes for q even. The case q odd was solved in [3]. For n≠3 we also determine their second smallest distance. Furthermore, we show that for q even all minimum weight codewords are equivalent and that the symplectic line-Grassmann codes are proper subcodes of codimension 2n of the orthogonal ones.

Subspace arrangements as generalized star configurations

AbstractIn this note, we show that any projective subspace arrangement can be realized as a generalized star configuration variety. This type of interpolation result may be useful in designing linear codes with prescribed codewords of minimum weight, as well as in answering a couple of questions about the number of equations needed to define a generalized star configuration variety.

Minimum distance and the minimum weight codewords of Schubert codes

We consider linear codes associated to Schubert varieties in Grassmannians. A formula for the minimum distance of these codes was conjectured in 2000 and after having been established in various special cases, it was proved in 2008 by Xiang. We give an alternative proof of this formula. Further, we propose a characterization of the minimum weight codewords of Schubert codes by introducing the notion of Schubert decomposable elements of certain exterior powers. It is shown that codewords corresponding to Schubert decomposable elements are of minimum weight and also that the converse is true in many cases. A lower bound, and in some cases, an exact formula, for the number of minimum weight codewords of Schubert codes is also given. From a geometric point of view, these results correspond to determining the maximum number of Fq-rational points that can lie on a hyperplane section of a Schubert variety in a Grassmannian with its nondegenerate embedding in a projective subspace of the Plücker projective space, and also the number of hyperplanes for which the maximum is attained.

Construction of parity‐check‐concatenated polar codes based on minimum Hamming weight codewords

In this Letter, a new parity-check-concatenated (PCC) polar code construction that considers the number of minimum Hamming weight (MHW) codewords is proposed. The parity bits to reduce the number of MHW codewords as much as possible is successively constructed. The proposed construction can reduce the number of MHW codewords more than other codes. The results show that the proposed codes can outperform the other codes at a high signal-to-noise region.

Weight Distributions of Non-Binary Multi-Edge Type LDPC Code Ensembles: Analysis and Efficient Evaluation

Non-binary multi-edge type ensembles of low-density parity-check codes are analyzed in terms of non-binary codeword weight distribution and its growth rate. In particular, an exact expression of the growth rate for small weights is developed. As a side result, the stopping set distributions of these ensembles are developed. Examples of weight distributions are provided, showing that the derived closed-form expressions can be easily evaluated. The obtained results can thus be exploited to analyze and design non-binary low-density parity-check codes that fall within the multi-edge type framework such as, but not limited to, protograph-based codes.

A Fast Method to Estimate Partial Weights Enumerators by Hash Techniques and Automorphism Group

BCH codes have high error correcting capability which allows classing them as good cyclic error correcting codes. This important characteristic is very useful in communication and data storage systems. Actually after almost 60 years passed from their discovery, their weights enumerators and therefore their analytical performances are known only for the lengths less than or equal to 127 and only for some codes of length as 255. The Partial Weights Enumerator (PWE) algorithm permits to obtain a partial weights enumerators for linear codes, it is based on the Multiple Impulse Method combined with a Monte Carlo Method; its main inconveniece is the relatively long run time. In this paper we present an improvement of PWE by integration of Hash techniques and a part of Automorphism Group (PWEHA) to accelerate it. The chosen approach applies to two levels. The first is to expand the sample which contains codewords of the same weight from a given codeword, this is done by adding a part of the Automorphism Group. The second level is to simplify the search in the sample by the use of hash techniques. PWEHA has allowed us to considerably reduce the run time of the PWE algorithm, for example that of PWEHA is reduced at more than 3900% for the BCH (127,71,19) code. This method is validated and it is used to approximate a partial weights enumerators of some BCH codes of unknown weights enumerators.

On the next-to-minimal weight of affine cartesian codes

In this paper we determine many values of the second least weight of codewords, also known as the next-to-minimal Hamming weight, for a type of affine variety codes, obtained by evaluating polynomials of degree up to d on the points of a cartesian product of n subsets of a finite field Fq. Such codes firstly appeared in a work by O. Geil and C. Thomsen (see [12]) as a special case of the so-called weighted Reed–Muller codes, and later appeared independently in a work by H. López, C. Rentería-Marquez and R. Villarreal (see [16]) named as affine cartesian codes. Our work extends, to affine cartesian codes, the results obtained by Rolland in [17] for generalized Reed–Muller codes.

Structural Analysis of Array-Based Non-Binary LDPC Codes

Structural properties of array-based non-binary low-density parity-check (NBLDPC) codes are studied in this paper. First, we characterize graphical substructures induced by codewords of symbol weight six in array-based NBLDPC codes defined by parity-check matrices with column weight three. We also reveal necessary conditions for these graphical substructures to incur weight-6 codewords. Such conditions can be used to select nonzero elements for avoiding weight-6 codewords or reducing the multiplicity of weight-6 codewords. Second, we show that there exist weight-7 codewords in array-based NBLDPC codes defined by parity-check matrices with column weight three. As a byproduct, we find that the graphical substructure induced by a weight-7 codeword takes the graphical substructure induced by the related weight-6 codewords as a subgraph. Third, we show that there may exist codewords with symbol weight four, six, and seven in array-based NBLDPC codes defined by parity-check matrices with column weight two. These results enrich the structural analysis of array-based LDPC codes. In addition, simulation results show the performance advantage of array-based NBLDPC codes.

Design and analysis of codes with distance 4 and 6 minimizing the probability of decoder error

The problem of minimization of the decoder error probability is considered for shortened codes of dimension 2m with distance 4 and 6. We prove that shortened Panchenko codes with distance 4 achieve the minimal probability of decoder error under special form of shortening. This shows that Hamming codes are not the best. In the paper, the rules for shortening Panchenko codes are defined and a combinatorial method to minimize the number of words of weight 4 and 5 is developed. There are obtained exact lower bounds on the probability of decoder error and the full solution of the problem of minimization of the decoder error probability for [39,32,4] and [72,64,4] codes. For shortened BCH codes with distance 6, upper and lower bounds on the number of minimal weight codewords are derived. There are constructed [45,32,6] and [79,64,6] BCH codes with the number of weight 6 codewords close to the lower bound and the decoder error probabilities are calculated for these codes. The results are intended for use in memory devices.

New Efficient Scheme Based on Reduction of the Dimension in the Multiple Impulse Method to Find the Minimum Distance of Linear Codes

In order to find a minimum weight codeword in a linear code, the Multiple Impulse Method uses the Ordered Statistics Decoder of order 3 having a complexity which increases with the code dimension. This paper presents an important improvement of this method by finding a sub code of C of small dimension containing a lowest weight codeword. In the case of Binary Extended Quadratic Residue codes, the proposed technique consists on finding a self invertible permutation ( from the projective special linear group and searching a codeword having the minimum weight in the sub code fixed by (. The proposed technique gives the exact value of the minimum distance for all binary quadratic residue codes of length less than 223 by using the Multiple Impulse Method on the sub codes in less than one second. For lengths more than 223, the obtained results prove the height capacity of the proposed technique to find the lowest weight in less time. The proposed idea is generalized for BCH codes and it has permits to find the true value of the minimum distance for some codes of lengths 1023 and 2047. The proposed methods performed very well in comparison to previously known results.

Estimates for distribution of the minimal distance of a random linear code

Abstract The distribution function of the minimum distance (the minimum weight of nonzero codewords) of a random linear code over a finite field is studied. Expicit bounds in the form of inequalities and asymptotic estimates for this distribution function are obtained.

Coding Schemes to Minimize Energy Consumption of Communication Links in Wireless Nanosensor Networks

It is critical to design energy-efficient communication technologies for wireless nanosensor networks (WNSNs) as nanosensors are highly energy-constrained. This paper focuses on WNSNs adopting the on – off keying (OOK) modulator. So far, some existing low-weight (LW) codes with low average codeword weight (ACW) map source symbols into different codewords with fewer high bits so as to greatly reduce the transmission energy at the transmitter. However, the transmission energy reduction is achieved at the price of large reception energy at the receiver as the codeword lengths of LW codes are large, incurring their ineffectiveness in most scenarios. To remedy the problem, we design the fixed-length minimum-communication-energy (F-MCE) code and variable-length minimum-communication-energy (V-MCE) code to minimize the total energy consumption at the transmitter and receiver for point-to-point communication in OOK-based WNSNs. Specifically, the code design problems are formulated as integer nonlinear programming (INLP) problems, and the F-MCE and V-MCE codes are obtained by solving the INLP problems. The F-MCE and V-MCE codes are applicable in more scenarios than the LW code. Extensive experimental results show that the V-MCE always outperforms the LW code regarding the energy saving, while the F-MCE code achieves energy saving no less than that of the existing LW codes.

Energy-efficient coding for electromagnetic nanonetworks in the Terahertz band

Wireless nanosensor networks (WNSNs) consist of numerous sensors with a nanoscale size and have many applications, such as the Internet of Nano-Things. The energy storage capacity of nanosensors is very limited; thus, energy efficiency has become an important issue in WNSNs. In this study, we investigate energy-efficient coding strategies and an energy consumption model for WNSNs. First, to map source words with equal probability of occurrence into different longer codewords, the average codeword weight minimization coding is presented. An energy consumption model based on energy consumption analysis is then proposed for WNSNs by jointly accounting for the energy consumption of both sender and receiver. Finally, the performance of the proposed coding scheme and energy consumption model is numerically and analytically investigated. Results show that the proposed coding is a promising scheme for WNSNs with a relatively low bit error rate.

Error-correction of linear codes via colon ideals

We show that errors in data transmitted through linear codes can be thought of as codewords of minimum weight of new linear codes. To determine errors we can then use methods specific to finding such special codewords. One of these methods consists of finding the primary decomposition of the saturation of a certain homogeneous ideal. When good words (i.e. vectors with a unique nearest neighbor) are error-corrected, the saturated ideal is just the prime ideal of a point (so the primary decomposition is superfluously determined); we show that this ideal can be computed by coloning the original homogeneous ideal with a power of a certain variable. We then determine the smallest such power for any linear code.

New Families of Optimal Variable-Weight Optical Orthogonal Codes With High Weights

The optical orthogonal codes (OOCs) have been widely used as spreading codes in communication systems employing the unipolar transmission. They are classified into constant-weight OOCs (CW-OOCs) and variable-weight OOCs (VW-OOCs) according to the number of distinct Hamming weights which their codewords have. In this paper, we present a new generic construction of VW-OOCs of length $(q-1)N$ from a CW-OOC of length $N$ , where $q$ is a prime power and $\gcd (q-1,N)=1$ . As a result, three new families of optimal VW-OOCs with a maximum correlation value 1 are obtained. In particular, these families can have the codewords of high weights, while most of the previously known optimal VW-OOCs have only codewords of weight less than 8.