Number Of Such Codes Research Articles

Hermitian self-dual 2-quasi-abelian codes

In this paper, we construct a class of Hermitian self-dual 2-quasi-abelian codes over a finite field. Based on counting the number of such codes and estimating the number of the codes in this class whose relative minimum weights are small, we prove that the class of Hermitian self-dual 2-quasi-abelian codes over any finite field is asymptotically good. The existence of such codes is unconditional, which is different from the case of Euclidean self-dual 2-quasi-abelian codes over a special finite field.

On the equivalence of Zps\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_{p^s}$$\\end{document}-linear generalized Hadamard codes

Linear codes of length n over Zps\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_{p^s}$$\\end{document}, p prime, called Zps\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_{p^s}$$\\end{document}-additive codes, can be seen as subgroups of Zpsn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_{p^s}^n$$\\end{document}. A Zps\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_{p^s}$$\\end{document}-linear generalized Hadamard (GH) code is a GH code over Zp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_p$$\\end{document} which is the image of a Zps\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_{p^s}$$\\end{document}-additive code under a generalized Gray map. It is known that the dimension of the kernel allows to classify these codes partially and to establish some lower and upper bounds on the number of such codes. Indeed, in this paper, for p≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 3$$\\end{document} prime, we establish that some Zps\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Z}_{p^s}$$\\end{document}-linear GH codes of length pt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^t$$\\end{document} having the same dimension of the kernel are equivalent to each other, once t is fixed. This allows us to improve the known upper bounds. Moreover, up to t=10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t=10$$\\end{document} if p=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=3$$\\end{document} or t=8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t=8$$\\end{document} if p=5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=5$$\\end{document}, this new upper bound coincides with a known lower bound based on the rank and dimension of the kernel.

Self-Dual 2-Quasi Abelian Codes

A class of self-dual quasi-abelian codes of index 2 over any finite field <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> is introduced. By counting the number of such codes and the number of the codes in this class whose relative minimum weights are small, such codes are proved to be asymptotically good provided −1 is a square in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> . Moreover, a class of self-orthogonal quasi-abelian codes of index 2 is defined; and such codes always exist. In a way similar to that for self-dual quasi-abelian codes of index 2, it is proved that these self-orthogonal quasi-abelian codes of index 2 are asymptotically good.

ПОСТРОЕНИЕ КРИТЕРИЯ ОЦЕНКИ КАЧЕСТВА ДВОИЧНЫХ БАРКЕРОПОДОБНЫХ КОДОВ

The objective of this article is to construct a criterion for evaluating the quality of binary Barker-like codes. The analysis of well-known studies on the formation of these codes with properties close to binary Barker codes is carried out. The analysis showed that in the works devoted to the use of well-known binary Barker-like codes in secure wireless communication systems and the formation of such codes, researchers primarily pay attention to such quality assessment indicators as the autocorrelation function, the length and number of such codes, and others. Based on the analysis of works [5, 9-25], the initial tuple of quality assessment of binary Barker-like codes is formulated, represented by the expression (1). However, the original tuple (1) lacks such an important quality assessment indicator as the linear complexity assessment, which is often taken as one of the primary indicators of crypto resistance. Based on this, the purpose of this article is to increase the accuracy of the evaluation of binary Barker-like codes by applying the identified criterion. The authors of the article, using the Berlekamp-Massey algorithm, evaluated the linear complexity of binary Barker-like codes presented in [9-23]. It has been established that a sufficiently large number of such codes (from 24% to 75%) have insufficient linear complexity, that is, they are non-cryptographic. As a result, they cannot be used in secure wireless communication systems. The result obtained is consistent with other studies in this field. Proceeding from this, when using well-known binary Barker-like codes in secure wireless communication systems, as well as when forming new binary Barker-like codes, it is necessary to pay attention not only to the known quality assessment indicators, but also to their linear complexity. The authors have modified the original tuple represented by expression (1). The modified tuple is represented by expression (2). According to the authors, the modified quality assessment criterion will improve the accuracy of the evaluation of binary Barker-like codes.

Constacyclic Codes over Finite Chain Rings of Characteristic p

Let R be a finite commutative chain ring of characteristic p with invariants p,r, and k. In this paper, we study λ-constacyclic codes of an arbitrary length N over R, where λ is a unit of R. We first reduce this to investigate constacyclic codes of length ps (N=n1ps,p∤n1) over a certain finite chain ring CR(uk,rb) of characteristic p, which is an extension of R. Then we use discrete Fourier transform (DFT) to construct an isomorphism γ between R[x]/&lt;xN−λ&gt; and a direct sum ⊕b∈IS(rb) of certain local rings, where I is the complete set of representatives of p-cyclotomic cosets modulo n1. By this isomorphism, all codes over R and their dual codes are obtained from the ideals of S(rb). In addition, we determine explicitly the inverse of γ so that the unique polynomial representations of λ-constacyclic codes may be calculated. Finally, for k=2 the exact number of such codes is provided.

On the structure of cube tiling codes

Let S be a set of arbitrary objects, and let Sd={v1...vd:vi∈S}. A polybox code is a set V⊂Sd with the property that for every two words v,w∈V there is i∈[d] with vi′=wi, where a permutation s↦s′ of S is such that s′′=(s′)′=s and s′≠s. If |V|=2d, then V is called a cube tiling code. Cube tiling codes determine 2-periodic cube tilings of Rd or, equivalently, tilings of the flat torus Td={(x1,…,xd)(mod2):(x1,…,xd)∈Rd} by translates of the unit cube as well as r-perfect codes in Z4r+2d in the maximum metric. By a structural result, cube tiling codes for d=4 are enumerated. It is computed that there are 27,385 non-isomorphic cube tiling codes in dimension four, and the total number of such codes is equal to 17,794,836,080,455,680. Moreover, some procedure of passing from a cube tiling code to a cube tiling code in dimensions d≤5 is given.

Equivalences among [formula omitted]-linear Hadamard codes

The Z2s-additive codes are subgroups of Z2sn, and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive code. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some Z2s-linear Hadamard codes of length 2t are equivalent, once t is fixed. This allows us to improve the known upper bounds for the number of such nonequivalent codes. Moreover, up to t=11, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel). Finally, when we focus on s∈{2,3}, the full classification of the Z2s-linear Hadamard codes of length 2t is established by giving the exact number of such codes.

Permutation-invariant qudit codes from polynomials

A permutation-invariant quantum code on N qudits is any subspace stabilized by the matrix representation of the symmetric group SN as permutation matrices that permute the underlying N subsystems. When each subsystem is a complex Euclidean space of dimension q≥2, any permutation-invariant code is a subspace of the symmetric subspace of (Cq)N. We give an algebraic construction of new families of d-dimensional permutation-invariant codes on at least (2t+1)2(d−1) qudits that can also correct t errors for d≥2. The construction of our codes relies on a real polynomial with multiple roots at the roots of unity, and a sequence of q−1 real polynomials that satisfy some combinatorial constraints. When N>(2t+1)2(d−1), we prove constructively that an uncountable number of such codes exist.

Efficient Search of Girth-Optimal QC-LDPC Codes

In this paper, we study the cycle structure of quasi-cyclic (QC) low-density parity-check (LDPC) codes with the goal of obtaining the shortest code with a given degree distribution and girth. We focus on QC-LDPC codes, whose Tanner graphs are cyclic liftings of fully connected base graphs of size 3 × n, n ≥ 4, and obtain minimal lifting degrees that result in girths 6 and 8. This is performed through an efficient exhaustive search, and as a result, we also find all the possible non-isomorphic codes with the same minimum block length, girth, and degree distribution. The exhaustive search, which is ordinarily a formidable task, is made possible by pruning the search space of many codes that are isomorphic to those previously examined in the search process. Many of the pruning techniques proposed in this paper are also applicable to QC-LDPC codes with base graphs other than the 3 × n fully connected ones discussed here, as well as to codes with a larger girth. To further demonstrate the effectiveness of the pruning techniques, we use them to search for QC-LDPC codes with girths 10 and 12, and find a number of such codes that have a shorter block length compared with the best known similar codes in the literature. In addition, motivated by the exhaustive search results, we tighten the lower bound on the block length of QC-LDPC codes of girth 6 constructed from fully connected 3 × n base graphs, and construct codes that achieve the lower bound for an arbitrary value of n ≥ 4.

Enumeration and construction of additive cyclic codes over Galois rings

Let R=GR(pϵ,l) be a Galois ring of characteristic pϵ and cardinality pϵl, where p and l are prime integers. First, we give a canonical form decomposition for additive cyclic codes over R. This decomposition is used to construct additive cyclic codes and count the number of such codes, respectively. Then we give the trace dual code for each additive cyclic code over R from its canonical form decomposition and linear codes of length l over some extension Galois rings of Zpϵ.

On cyclic self-orthogonal codes over [formula omitted

In this paper, we study cyclic self-orthogonal codes over Z2m for any odd length. We give the generator polynomials of cyclic self-orthogonal codes over Z2m. By using these generator polynomials, we obtain a sufficient and necessary condition for the existence of nontrivial cyclic self-orthogonal codes over Z2m and determine the number of such codes for a given odd length. Some results about cyclic self-dual codes over Z2m are included.

A note on cyclic codes over GR(p 2, m) of length p k

The number of self-dual cyclic codes of length p k over GR(p 2, m) is determined by the nullity of a certain matrix M(p k , i 1). With the aid of Genocchi numbers, we determine the nullity of M(p k , i 1) and hence determine completely the number of such codes.

The number of self-dual codes over $${Z_{p^3}}$$

Let p be a prime number. In this paper, we consider codes over the ring $${Z_{p^3}}$$ of integers modulo p 3 and give a characterization of self-duality. This leads to a construction of self-dual codes and a mass formula, which counts the number of such codes over $${Z_{p^3}}$$ .

On the Number of $1$-Perfect Binary Codes: A Lower Bound

We present a construction of 1-perfect binary codes, which gives a new lower bound on the number of such codes. We conjecture that this lower bound is asymptotically tight.

Additive cyclic codes over $\mathbb F_4$

In this paper we find a canonical form decomposition for additive cyclic codes of even length over $\mathbb F_4$. This decomposition is used to count the number of such codes. We also prove that each code is the $\mathbb F_2$-span of at most two codewords and their cyclic shifts. We examine the construction of additive cyclic self-dual codes of even length and apply these results to those codes of length 24.

A new class of interlaced complementary codes based on components with unity peak sidelobes

A new class of biphase complementary code sets is proposed. Individual codes in each set have maximum sidelobes of unity magnitude. The individual codes could have gaps of zeros but they are interlaced together without gaps in the final scheme. A number of such codes, as presented in this paper, use Barker codes as building blocks with additional elements from { + 1 ,- 1 , 0 } . The main drawback of regular complementary codes longer than 4 is that they have peak sidelobe magnitude greater than unity. In the presence of frequency selective fading or other factors, inexact sidelobe cancellation results in non-zero sidelobes at the output. This is minimized in the proposed schemes by using codes that have peak sidelobes of unity magnitude as the individual codes in the sets. The proposed codes are attractive due to their increased resistance to the effect of inexact sidelobe cancellation. A figure of merit is proposed to measure the frequency use efficiency of the proposed codes.

Additive cyclic codes over $\mathbb F_4$

In this paper we find a canonical form decomposition for additive cyclic codes of odd length over $\mathbb F_4$. This decomposition is used to count the number of such codes. We also reprove that each code is the $\mathbb F_2$-span of at most two codewords and their cyclic shifts, a fact first proved in [2]. A count is given for the number of codes that are the $\mathbb F_2$-span of one codeword and its cyclic shifts. We can examine this decomposition to see precisely when the code is self-orthogonal or self-dual under the trace inner product. Using this, a count is presented for the number of self-orthogonal and self-dual additive cyclic codes of odd length. We also provide a count of the additive cyclic and additive cyclic self-orthogonal codes as a function of their $\mathbb F_2$-dimension.

Enumeration of inequivalent irreducible Goppa codes

We consider irreducible Goppa codes over F q of length q n defined by polynomials of degree r, where q is a prime power and n , r are arbitrary positive integers. We obtain an upper bound on the number of such codes.

An Information-Theoretic View of Network Management

We present an information-theoretic framework for network management for recovery from nonergodic link failures. Building on recent work in the field of network coding, we describe the input-output relations of network nodes in terms of network codes. This very general concept of network behavior as a code provides a way to quantify essential management information as that needed to switch among different codes (behaviors) for different failure scenarios. We compare two types of recovery schemes, receiver-based and network-wide, and consider two formulations for quantifying network management. The first is a centralized formulation where network behavior is described by an overall code determining the behavior of every node, and the management requirement is taken as the logarithm of the number of such codes that the network may switch among. For this formulation, we give bounds, many of which are tight, on management requirements for various network connection problems in terms of basic parameters such as the number of source processes and the number of links in a minimum source-receiver cut. Our results include a lower bound for arbitrary connections and an upper bound for multitransmitter multicast connections, for linear receiver-based and network-wide recovery from all single link failures. The second is a node-based formulation where the management requirement is taken as the sum over all nodes of the logarithm of the number of different behaviors for each node. We show that the minimum node-based requirement for failures of links adjacent to a single receiver is achieved with receiver-based schemes.

Abelian codes over galois rings closed under certain permutations

We study n-length Abelian codes over Galois rings with characteristic p/sup a/, where n and p are relatively prime, having the additional structure of being closed under the following two permutations: (i) permutation effected by multiplying the coordinates with a unit in the appropriate mixed-radix representation of the coordinate positions and (ii) shifting the coordinates by t positions. A code is t-quasi-cyclic (t-QC) if t is an integer such that cyclic shift of a codeword by t positions gives another codeword. We call the Abelian codes closed under the first permutation as unit-invariant Abelian codes and those closed under the second as quasi-cyclic Abelian (QCA) codes. Using a generalized discrete Fourier transform (GDFT) defined over an appropriate extension of the Galois ring, we show that unit-invariant Abelian and QCA codes can be easily characterized in the transform domain. For t=1, QCA codes coincide with those that are cyclic as well as Abelian. The number of such codes for a specified size and length is obtained and we also show that the dual of an unit-invariant t-QCA code is also an unit-invariant t-QCA code. Unit-invariant Abelian (hence unit-invariant cyclic) and t-QCA codes over Galois field F/sub p//sup l/ and over the integer residue rings are obtainable as special cases.