Q-ary Codes Research Articles

Bounds on the Probability of Undetected Error for q-Ary Codes.

We study the probability of an undetected error for general q-ary codes. We give upper and lower bounds on this quantity, by the Linear Programming and the Polynomial methods, as a function of the length, size, and minimum distance. Sharper bounds are obtained in the important special case of binary Hamming codes. Finally, several examples are given to illustrate the results of this paper.

Construction of quasi-cyclic self-dual codes over finite fields

ABSTRACT Our goal of this paper is to find a construction of all ℓ-quasi-cyclic self-dual codes over a finite field F q of length mℓ for every positive even integer ℓ. In this paper, we study the case where x m − 1 has an arbitrary number of irreducible factors in F q [ x ] ; in the previous studies, only some special cases where x m − 1 has exactly two or three irreducible factors in F q [ x ] , were studied. Firstly, the binary code case is completed: for any even positive integer ℓ, every binary ℓ-quasi-cyclic self-dual code can be obtained by our construction. Secondly, we work on the q-ary code cases for an odd prime power q. We find an explicit method for construction of all ℓ-quasi-cyclic self-dual codes over F q of length mℓ for any even positive integer ℓ, where we require that q ≡ 1 ( mod 4 ) if the index ℓ ≥ 6 . By implementation of our method, we obtain a new optimal binary self-dual code [ 172 , 86 , 24 ] , which is also a quasi-cyclic code of index 4.

On the generalization of the construction of quantum codes from Hermitian self-orthogonal codes

Many q-ary stabilizer quantum codes can be constructed from Hermitian self-orthogonal q^2-ary linear codes. This result can be generalized to q^{2 m}-ary linear codes, m > 1. We give a result for easily obtaining quantum codes from that generalization. As a consequence we provide several new binary stabilizer quantum codes which are records according to Grassl (Bounds on the minimum distance of linear codes, http://www.codetables.de, 2020) and new q-ary ones, with q ne 2, improving others in the literature.

On the Global and Local Decoding of q-Ary SC-LDPC Codes With Moderate Coupling Length

This letter discusses a special type of q-ary spatially coupled low-density parity-check (SC-LDPC) codes with the potential for local decoding (LD). We demonstrate that q-ary SC-LDPCL (suffix “L” stands for locality) code ensembles have better decoding thresholds than the corresponding binary ensembles. We also develop local window decoding (LWD) to improve local decoding performance. The q-ary codes are constructed from low-order finite fields that guarantees relatively low complexity. The provided examples show that q-ary SC-LDPCL codes outperform the corresponding binary codes in both global and local decoding.

An Embedding Strategy Using Q-Ary Convolutional Codes for Large and Small Payloads

Matrix embedding (ME) code is a commonly used steganography technique, which uses linear block codes to improve embedding efficiency. However, its main disadvantage is the inability to perform maximum likelihood decoding due to the high complexity of decoding large ME codes. As such, it is difficult to improve the embedding efficiency. The proposed q-ary embedding code can provide excellent embedding efficiency and is suitable for various embedding rates (large and small payloads). This article discusses that by using perforation technology, a convolutional code with a high embedding rate can be easily converted into a convolutional code with a low embedding rate. By keeping the embedding rate of the (2, 1) convolutional code unchanged, convolutional codes with different embedding rates can be designed through puncturing.

On homogeneous arcs and linear codes over finite chain rings

We consider projective Hjelmslev geometries over finite chain rings of length 2 with residue field of order q. In these geometries we introduce and investigate the so-called homogeneous arcs defined as multisets of points with respect to a special weight function. These arcs are associated with linearly representable q-ary codes. We establish a relation between the parameters of a linearly representable code and the parameters of the associated arc. We prove an inequality for the largest homogeneous weight of an arc of given size N. We consider the $$\tau $$ -dual of a homogeneous arc and give an explicit formula for the homogeneous weights of the hyperplanes with respect to the dual arc. We prove that if a homogeneous arc is of constant weight, this weight is forced to be zero and the arc is a sum of neighbour classes of points. We prove various numerical conditions on the parameters of homogeneous two-weight arcs, as well as an interesting identity involving the parameters of a projective homogeneous two-weight arc and its dual, which isturns out to be also a homogeneous two-weight arc. Finally, we present examples of two-weight homogeneous arcs some of which are new.

A New Class of Q-Ary Codes for the McEliece Cryptosystem

The McEliece cryptosystem is a promising candidate for post-quantum public-key encryption. In this work, we propose q-ary codes over Gaussian integers for the McEliece system and a new channel model. With this one Mannheim error channel, errors are limited to weight one. We investigate the channel capacity of this channel and discuss its relation to the McEliece system. The proposed codes are based on a simple product code construction and have a low complexity decoding algorithm. For the one Mannheim error channel, these codes achieve a higher error correction capability than maximum distance separable codes with bounded minimum distance decoding. This improves the work factor regarding decoding attacks based on information-set decoding.

Efficient Encoding/Decoding of GC-Balanced Codes Correcting Tandem Duplications

Tandem duplication is the process of inserting a copy of a segment of DNA adjacent to the original position. Motivated by applications that store data in living organisms, Jain et al. (2017) proposed the study of codes that correct tandem duplications. All code constructions are based on irreducible words. Such code constructions are almost optimal to combat tandem duplications of length at most k where k ≤ 3. However, the problem of designing efficient encoder/decoder for such codes has not been investigated. In addition, the method cannot be extended to deal with the case of arbitrary k, where k ≥ 4. In this work, we study efficient encoding/decoding methods for irreducible words over general q-ary alphabet. Our methods provide the first known efficient encoder/decoder for q-ary codes correcting tandem duplications of length at most k, where k ≤ 3. In particular, we describe an (1, m)-finite state encoder and show that when m = Θ(1/ε) and ϊ = Θ(1/ε), the encoder achieves rate that is ε away from the optimal rate. We also provide ranking/unranking algorithms for irreducible words and modify the algorithms to reduce the space requirements for the finite state encoder. Over the DNA alphabet (or quaternary alphabet), we also impose weight constraint on the codewords. In particular, a quaternary word is GC-balanced if exactly half of the symbols of are either C or G. Via a modification of Knuth's balancing technique, we provide an efficient method that translates quaternary messages into GC-balanced codewords and the resulting codebook is able to correct tandem duplications of length at most k, where k ≤ 3. In addition, we provide the first known construction of codes to combat tandem duplications of length at most k, where k ≥ 4. Such codes can correct duplication errors in linear-time and they are almost optimal in terms of rate.

New Bounds on 2-Frameproof Codes of Length 4

Frameproof codes were first introduced by Boneh and Shaw in 1998 in the context of digital fingerprinting to protect copyrighted materials. These digital fingerprints are generally denoted as codewords in Qn, where Q is an alphabet of size q and n is a positive integer. A 2-frameproof code is a code C such that any 2 codewords in C cannot form a new codeword under a particular rule. Thus, no pair of users can frame a user who is not a member of the coalition. This paper concentrates on the upper bound for the size of a q-ary 2-frameproof code of length 4. Our new upper bound shows that C≤2q2−2q+1 when q is odd and q>10.

On q-ary Codes with Two Distances d and d + 1

The $q$-ary block codes with two distances $d$ and $d+1$ are considered. Several constructions of such codes are given, as in the linear case all codes can be obtained by a simple modification of linear equidistant codes. Upper bounds for the maximum cardinality of such codes is derived. Tables of lower and upper bounds for small $q$ and $n$ are presented.

Multi-level QR code with Hill Climbed Encrypted Public Level for Secured Message Transfer

With the world turning into a worldwide town, the data sharing requires abnormal state security with substantial verification of the data which ought not be undermined with the time, hence providing food all the above necessities we propose another brisk reaction (QR) code containing two stockpiling levels specifically open and private stockpiling levels, which can go above and beyond in validating the inventiveness of the reports. The open stockpiling level can be perused by any traditional QR code scanners while it's not the equivalent for the private stockpiling level as its being produced with exchanging the dark modules with finished examples of the first QR code consequently it tends to be perused by just explicit QR scanners. The data's being put away in the private stockpiling level is being encoded sing q-ary code substituted by mistake amendment systems along these lines expanding the QR stockpiling limit while the open stockpiling level data's are being scrambled with the key which will be sent to the private stockpiling level where it experiences all the procedure like the private data's and will be moved by means of QR to the required gatherings. The size can be expanded by the information's or with the finished example estimate

Two-weight codes: Upper bounds and new optimal constructions

In this paper, we consider q-ary codes of length n and minimum Hamming distance d, which have two weights w1 and w2. These codes are denoted by (n,d,{w1,w2})q codes. Let Aq(n,d,{w1,w2}) denote the largest possible number of codewords in an (n,d,{w1,w2})q code and we simply write A(n,d,{w1,w2}) for A2(n,d,{w1,w2}). Some upper bounds on Aq(n,d,{w1,w2}) are given. The equivalences between binarytwo-weight codes and special combinatorial configurations with certain properties are established and then new upper bounds on A(n,d,{w1,w2}) are derived. For w1,w2∈{2,3,4}, optimal constructions of (n,d,{w1,w2})2 codes are presented. The exact value of A(n,d,{2,3}) is completely determined for all n and d. We determine the exact value of A(n,d,{2,4}) for any positive integer n≡2,4(mod6) and d∈{3,4}. The exact value of A(n,d,{3,4}) is determined for any integer n and d∈{6,7}, or d=5 and n≡0,2,3,11(mod12).

Some q-ary Cyclic Codes from Explicit Monomials over $$\mathbb{F}_{q}m$$

Cyclic codes as a subclass of linear codes have practical applications in communication systems, consumer electronics, and data storage systems due to their efficient encoding and decoding algorithms. The objective of this paper is to construct some cyclic codes by the sequence approach. More precisely, we determine the dimension and the generator polynomials of three classes of q-ary cyclic codes defined by some sequences with explicit polynomials over \(\mathbb{F}_{q}m\). The minimum distance of such cyclic codes is also discussed. Some of these codes are optimal according to code tables. Moreover, the third class of cyclic codes provides some answers for Open Problem 3 proposed by Ding and Zhou in [1].

High Rate Robust Codes with Low Implementation Complexity

Robust codes ${\mathcal {C}}(n,k)_q$C(n,k)q are nonlinear $q$q-ary codes of dimension $k$k and length $n\leq 2k$n≤2k. Robust codes can detect any error with nonzero probability; hence, they can effectively detect fault injection attacks. Most high rate robust codes are either restricted to certain ratios between $n$n and $k$k, or have relatively high hardware complexity. This paper presents new constructions for optimum or close to optimum low complexity high rate robust codes. These codes exist for any $k$k and $n$n. The hardware complexity of each construction is discussed, and a method to choose the one with the smallest implementation cost is presented.

On the third greedy weight of 4-dimensional codes

The upper bounds on the difference between the third greedy weight and the third generalized Hamming weight of 4-dimensional q-ary codes are obtained by using the finite geometry method. The codes achieving the upper bounds are constructed, and these codes are optimal with respect to the security when they are used in the wire-tap channel of type II with the coset coding scheme.

New Construction of Maximum Distance Separable (MDS) Self-Dual Codes over Finite Fields.

Maximum distance separable (MDS) self-dual codes have useful properties due to their optimality with respect to the Singleton bound and its self-duality. MDS self-dual codes are completely determined by the length so the problem of constructing q-ary MDS self-dual codes with various lengths is a very interesting topic. Recently X. Fang et al. using a method given in previous research, where several classes of new MDS self-dual codes were constructed through (extended) generalized Reed-Solomon codes, in this paper, based on the method given in we achieve several classes of MDS self-dual codes.

A new class of security oriented error correcting robust codes

Robust codes are codes that can detect any nonzero errore with probability 1 − Q(e) > 0. This property makes them useful in protecting hardware systems from fault injection attacks which cause an arbitrary number of bit flips. This paper presents a new construction of non-linear robust q-ary codes with q = 2m and an error correction capability. The codes are built upon systematic linear codes [n, k, d]q whereas the n − k redundant symbols that were originally allocated to increase the minimum distance of the code are modified to provide both correction capability and robustness. The error masking probability of the codes is Q(e) upper bounded by 2/q for odd values of m and by 4/q for even m. Hence, they are more effective in detecting maliciously injected errors and have a higher code rate than codes obtained by concatenation of a linear error correcting code with a security oriented code.

Linear codes close to the Griesmer bound and the related geometric structures

In this paper, we study the behavior of the function $$t_q(k)$$ defined as the maximal deviation from the Griesmer bound of the optimal length of a linear code with a fixed dimension k: $$\begin{aligned} t_q(k)=\max _d(n_q(k,d)-g_q(k,d)), \end{aligned}$$ where the maximum is taken over all minimum distances d. Here $$n_q(k,d)$$ is the shortest length of a q-ary linear code of dimension k and minimum distance d, $$g_q(k,d)$$ is the Griesmer bound for a code of dimension k and minimum distance d. We give two equivalent geometric versions of this problem in terms of arcs and minihypers. We prove that $$t_q(k)\rightarrow \infty $$ when $$k\rightarrow \infty $$ which implies that the problem is non-trivial. We prove upper bounds on the function $$t_q(k)$$ . For codes of even dimension k we show that $$t_q(k)\le 2(q^{k/2}-1)/(q-1)-(k+q-1)$$ which implies that $$t_q(k)\in O(q^{k/2})$$ for all k. For three-dimensional codes and q even we prove the stronger estimate $$t_q(3)\le \log q-1$$ , as well as $$t_q(3)\le \sqrt{q}-1$$ for the case q square.

On m-Near-Resolvable Block Designs and q-ary Constant-Weight Codes

We introduce m-near-resolvable block designs. We establish a correspondence between such block designs and a subclass of (optimal equidistant) q-ary constant-weight codes meeting the Johnson bound. We present constructions of m-near-resolvable block designs, in particular based on Steiner systems and super-simple t-designs.

A construction of q-ary linear codes with irreducible cyclic codes

Linear codes are an important class of error-correcting codes and widely used in secret sharing schemes, combinational designs, authentication codes and so on. The objective of this paper is to construct q-ary linear codes with good parameters from irreducible cyclic codes. Several classes of linear codes with a few weights including infinite families of distance-optimal ones are derived. The nonzero codewords of some of the codes in this paper have minimal support for inclusion and such codes can be used to construct secret sharing schemes with interesting access structures.